algebraic extension

Algebraic extensionFrom Wikipedia, the free encyclopedia

Contents

1 Abelian extension 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Algebraic closure 22.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Existence of an algebraic closure and splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Separable closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Algebraic element 43.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Algebraic extension 64.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Algebraic number eld 85.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.1.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.3 Algebraicity and ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.3.1 Unique factorization and class number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3.2 -functions, L-functions and class number formula . . . . . . . . . . . . . . . . . . . . . . 10

5.4 Bases for number elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4.1 Integral basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4.2 Power basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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5.5 Regular representation, trace and determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.6 Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6.1 Archimedean places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6.2 Nonarchimedean or ultrametric places . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6.3 Prime ideals in OF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.7 Ramication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.7.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.7.2 Dedekind discriminant theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.8 Galois groups and Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.9 Local-global principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.9.1 Local and global elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.9.2 Hasse principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.9.3 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Characteristic (algebra) 196.1 Other equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Case of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 Case of elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Countable set 227.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Formal denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.5 Minimal model of set theory is countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.6 Total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8 Degree of a eld extension 308.1 Denition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2 The multiplicativity formula for degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.2.1 Proof of the multiplicativity formula in the nite case . . . . . . . . . . . . . . . . . . . . 318.2.2 Proof of the formula in the innite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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8.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9 Dual basis in a eld extension 33

10 Field extension 3410.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.5 Algebraic and transcendental elements and extensions . . . . . . . . . . . . . . . . . . . . . . . . 3510.6 Normal, separable and Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.8 Extension of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Finite eld 3811.1 Denitions, rst examples, and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3 Explicit construction of nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11.3.1 Non-prime elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3.2 Field with four elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.3.3 GF(p2) for an odd prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.3.4 GF(8) and GF(27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.3.5 GF(16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11.4 Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4.1 Discrete logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

11.5 Frobenius automorphism and Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.6 Polynomial factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

11.6.1 Irreducible polynomials of a given degree . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.6.2 Number of monic irreducible polynomials of a given degree over a nite eld . . . . . . . . 45

11.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

11.8.1 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.8.2 Wedderburns little theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

11.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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11.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4711.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12 Galois extension 4812.1 Characterization of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13 Normal extension 5013.1 Equivalent properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.3 Normal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

14 Ring homomorphism 5214.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.3 The category of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

14.3.1 Endomorphisms, isomorphisms, and automorphisms . . . . . . . . . . . . . . . . . . . . . 5414.3.2 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

15 Root of unity 5515.1 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.2 Elementary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5715.4 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6115.5 Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6115.6 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6215.7 Cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6315.8 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6415.9 Cyclotomic elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.10Relation to integer rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.14Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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16.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6816.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6916.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6916.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.5 The denition of separable non-algebraic extension elds . . . . . . . . . . . . . . . . . . . . . . 7016.6 Dierential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

17 Simple extension 7317.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7317.2 Structure of simple extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7317.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7417.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

18 Splitting eld 7518.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.2 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3 Constructing splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

18.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3.2 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.3.3 The eld Ki[X]/(f(X)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

18.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.4.1 The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.4.2 Cubic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.4.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

18.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7818.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7818.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

19 Tower of elds 7919.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

20 Zorns lemma 8020.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8020.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8020.3 Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.5 Equivalent forms of Zorns lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.6 Pop Culture References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

vi CONTENTS

20.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.10Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 84

20.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8420.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8620.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Chapter 1

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois groupis a cyclic group, we have a cyclic extension. A Galois extension is called solvable if its Galois group is solvable, i.e.if it is constructed from a series of abelian groups corresponding to intermediate extensions.Every nite extension of a nite eld is a cyclic extension. The development of class eld theory has provided detailedinformation about abelian extensions of number elds, function elds of algebraic curves over nite elds, and localelds.There are two slightly dierent concepts of cyclotomic extensions: these can mean either extensions formed byadjoining roots of unity, or subextensions of such extensions. The cyclotomic elds are examples. Any cyclotomicextension (for either denition) is abelian.If a eld K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resultingso-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n,since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th rootsof elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-directproduct. The Kummer theory gives a complete description of the abelian extension case, and the KroneckerWebertheorem tells us that if K is the eld of rational numbers, an extension is abelian if and only if it is a subeld of a eldobtained by adjoining a root of unity.There is an important analogy with the fundamental group in topology, which classies all covering spaces of a space:abelian covers are classied by its abelianisation which relates directly to the rst homology group.

1.1 References Kuz'min, L.V. (2001), cyclotomic extension, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

1

Chapter 2

Algebraic closure

For other uses, see Closure (disambiguation).

In mathematics, particularly abstract algebra, an algebraic closure of a eld K is an algebraic extension of K that isalgebraically closed. It is one of many closures in mathematics.Using Zorns lemma, it can be shown that every eld has an algebraic closure,[1][2][3] and that the algebraic closureof a eld K is unique up to an isomorphism that xes every member of K. Because of this essential uniqueness, weoften speak of the algebraic closure of K, rather than an algebraic closure of K.The algebraic closure of a eld K can be thought of as the largest algebraic extension of K. To see this, note that if Lis any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is containedwithin the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed eld containingK, because ifM is any algebraically closed eld containing K, then the elements ofM that are algebraic over K forman algebraic closure of K.The algebraic closure of a eld K has the same cardinality as K if K is innite, and is countably innite if K is nite.[3]

2.1 Examples The fundamental theorem of algebra states that the algebraic closure of the eld of real numbers is the eld ofcomplex numbers.

The algebraic closure of the eld of rational numbers is the eld of algebraic numbers.

There are many countable algebraically closed elds within the complex numbers, and strictly containing theeld of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers,e.g. the algebraic closure of Q().

For a nite eld of prime power order q, the algebraic closure is a countably innite eld that contains a copyof the eld of order qn for each positive integer n (and is in fact the union of these copies).[4]

2.2 Existence of an algebraic closure and splitting eldsLet S = ffj 2 g be the set of all monic irreducible polynomials in K[x]. For each f 2 S , introduce newvariables u;1; : : : ; u;d where d = degree(f) . Let R be the polynomial ring over K generated by u;i for all 2 and all i degree(f) . Write

f dY

i=1

(x u;i) =d1Xj=0

r;j xj 2 R[x]

2

2.3. SEPARABLE CLOSURE 3

with r;j 2 R . Let I be the ideal in R generated by the r;j . By Zorns lemma, there exists a maximal idealM in Rthat contains I. Now R/M is an algebraic closure of K; every f splits as the product of the x (u;i +M) .The same proof also shows that for any subset S of K[x], there exists a splitting eld of S over K.

2.3 Separable closureAn algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separableextensions ofK withinKalg. This subextension is called a separable closure ofK. Since a separable extension of a sep-arable extension is again separable, there are no nite separable extensions of Ksep, of degree > 1. Saying this anotherway, K is contained in a separably-closed algebraic extension eld. It is essentially unique (up to isomorphism).[5]

The separable closure is the full algebraic closure if and only if K is a perfect eld. For example, if K is a eld ofcharacteristic p and if X is transcendental over K,K(X)( p

pX) K(X) is a non-separable algebraic eld extension.

In general, the absolute Galois group of K is the Galois group of Ksep over K.[6]

2.4 See also Algebraically closed eld Algebraic extension Puiseux expansion

2.5 References[1] McCarthy (1991) p.21

[2] M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp.11-12.

[3] Kaplansky (1972) pp.74-76

[4] Brawley, Joel V.; Schnibben, George E. (1989), 2.2 The Algebraic Closure of a Finite Field, Innite Algebraic Extensionsof Finite Fields, Contemporary Mathematics 95, American Mathematical Society, pp. 2223, ISBN 978-0-8218-5428-0,Zbl 0674.12009.

[5] McCarthy (1991) p.22

[6] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.

Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. ISBN 0-226-42451-0. Zbl 1001.16500.

McCarthy, Paul J. (1991). Algebraic extensions of elds (Corrected reprint of the 2nd ed.). New York: DoverPublications. Zbl 0768.12001.

Chapter 3

Algebraic element

In mathematics, if L is a eld extension of K, then an element a of L is called an algebraic element over K, or justalgebraic over K, if there exists some non-zero polynomial g(x) with coecients in K such that g(a)=0. Elements ofL which are not algebraic over K are called transcendental over K.These notions generalize the algebraic numbers and the transcendental numbers (where the eld extension is C/Q, Cbeing the eld of complex numbers and Q being the eld of rational numbers).

3.1 Examples The square root of 2 is algebraic over Q, since it is the root of the polynomial g(x) = x2 - 2 whose coecientsare rational.

Pi is transcendental over Q but algebraic over the eld of real numbers R: it is the root of g(x) = x - , whosecoecients (1 and -) are both real, but not of any polynomial with only rational coecients. (The denitionof the term transcendental number uses C/Q, not C/R.)

3.2 PropertiesThe following conditions are equivalent for an element a of L:

a is algebraic over K the eld extension K(a)/K has nite degree, i.e. the dimension of K(a) as a K-vector space is nite. (HereK(a) denotes the smallest subeld of L containing K and a)

K[a] = K(a), where K[a] is the set of all elements of L that can be written in the form g(a) with a polynomialg whose coecients lie in K.

This characterization can be used to show that the sum, dierence, product and quotient of algebraic elements overK are again algebraic over K. The set of all elements of L which are algebraic over K is a eld that sits in between Land K.If a is algebraic over K, then there are many non-zero polynomials g(x) with coecients in K such that g(a) = 0.However there is a single one with smallest degree and with leading coecient 1. This is the minimal polynomial ofa and it encodes many important properties of a.Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed.The eld of complex numbers is an example.

3.3 See also Algebraic independence

4

3.4. REFERENCES 5

3.4 References Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

Chapter 4

Algebraic extension

In abstract algebra, a eld extension L/K is called algebraic if every element of L is algebraic over K, i.e. if everyelement of L is a root of some non-zero polynomial with coecients in K. Field extensions that are not algebraic, i.e.which contain transcendental elements, are called transcendental.For example, the eld extensionR/Q, that is the eld of real numbers as an extension of the eld of rational numbers,is transcendental, while the eld extensionsC/R andQ(2)/Q are algebraic, whereC is the eld of complex numbers.All transcendental extensions are of innite degree. This in turn implies that all nite extensions are algebraic.[1] Theconverse is not true however: there are innite extensions which are algebraic. For instance, the eld of all algebraicnumbers is an innite algebraic extension of the rational numbers.If a is algebraic over K, then K[a], the set of all polynomials in a with coecients in K, is not only a ring but a eld:an algebraic extension of K which has nite degree over K. The converse is true as well, if K[a] is a eld, then a isalgebraic over K. In the special case where K =Q is the eld of rational numbers, Q[a] is an example of an algebraicnumber eld.A eld with no nontrivial algebraic extensions is called algebraically closed. An example is the eld of complexnumbers. Every eld has an algebraic extension which is algebraically closed (called its algebraic closure), but provingthis in general requires some form of the axiom of choice.An extension L/K is algebraic if and only if every sub K-algebra of L is a eld.

4.1 PropertiesThe class of algebraic extensions forms a distinguished class of eld extensions, that is, the following three propertieshold:[2]

1. If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.

2. If E and F are algebraic extensions of K in a common overeld C, then the compositum EF is an algebraicextension of K.

3. If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K.

These nitary results can be generalized using transnite induction:

1. The union of any chain of algebraic extensions over a base eld is itself an algebraic extension over the samebase eld.

This fact, together with Zorns lemma (applied to an appropriately chosen poset), establishes the existence of algebraicclosures.

6

4.2. GENERALIZATIONS 7

4.2 GeneralizationsMain article: Substructure

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding ofM into N is calledan algebraic extension if for every x in N there is a formula p with parameters inM, such that p(x) is true and the set

ny 2 N

p(y)ois nite. It turns out that applying this denition to the theory of elds gives the usual denition of algebraic extension.The Galois group of N overM can again be dened as the group of automorphisms, and it turns out that most of thetheory of Galois groups can be developed for the general case.

4.3 See also Integral element Lroths theorem Galois extension Separable extension Normal extension

4.4 Notes[1] See also Hazewinkel et al. (2004), p. 3.

[2] Lang (2002) p.228

4.5 References Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhalovna; Kirichenko, Vladimir V. (2004),Algebras, rings and modules 1, Springer, ISBN 1-4020-2690-0

Lang, Serge (1993), V.1:Algebraic Extensions, Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub.Co., pp. 223, ISBN 978-0-201-55540-0, Zbl 0848.13001

McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of elds, New York:Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001

Roman, Steven (1995), Field Theory, GTM 158, Springer-Verlag, ISBN 9780387944081 Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 9780130878687

Chapter 5

Algebraic number eld

In mathematics, an algebraic number eld (or simply number eld) F is a nite degree (and hence algebraic) eldextension of the eld of rational numbersQ. Thus F is a eld that containsQ and has nite dimension when consideredas a vector space over Q.The study of algebraic number elds, and, more generally, of algebraic extensions of the eld of rational numbers, isthe central topic of algebraic number theory.

5.1 Denition

5.1.1 Prerequisites

Main articles: Field and Vector space

The notion of algebraic number eld relies on the concept of a eld. A eld consists of a set of elements togetherwith two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent exampleof a eld is the eld of rational numbers, commonly denoted Q, together with its usual operations of addition etc.Another notion needed to dene algebraic number elds is vector spaces. To the extent needed here, vector spacescan be thought of as consisting of sequences (or tuples)

(x1, x2, ...)

whose entries are elements of a xed eld, such as the eld Q. Any two such sequences can be added by adding theentries one per one. Furthermore, any sequence can be multiplied by a single element c of the xed eld. These twooperations known as vector addition and scalar multiplication satisfy a number of properties that serve to dene vectorspaces abstractly. Vector spaces are allowed to be innite-dimensional, that is to say that the sequences constitutingthe vector spaces are of innite length. If, however, the vector space consists of nite sequences

(x1, x2, ..., xn),

the vector space is said to be of nite dimension, n.

5.1.2 Denition

An algebraic number eld (or simply number eld) is a nite degree eld extension of the eld of rational numbers.Here its dimension as a vector space over Q is simply called its degree.

8

5.2. EXAMPLES 9

5.2 Examples The smallest and most basic number eld is the eldQ of rational numbers. Many properties of general numberelds, such as unique factorization, are modelled after the properties of Q.

The Gaussian rationals, denoted Q(i) (read as "Q adjoined i"), form the rst nontrivial example of a numbereld. Its elements are expressions of the form

a+bi

where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added,subtracted, andmultiplied according to the usual rules of arithmetic and then simplied using the identity

i2 = 1.

Explicitly,

(a + bi) + (c + di) = (a + c) + (b + d)i,(a + bi) (c + di) = (ac bd) + (ad + bc)i.

Non-zero Gaussian rational numbers are invertible, which can be seen from the identity

(a+ bi)

a

a2 + b2 ba2 + b2

i

=

(a+ bi)(a bi)a2 + b2

= 1:

It follows that the Gaussian rationals form a number eld which is two-dimensional as a vector spaceover Q.

More generally, for any square-free integer d, the quadratic eld

Q(d)

is a number eld obtained by adjoining the square root of d to the eld of rational numbers. Arithmeticoperations in this eld are dened in analogy with the case of gaussian rational numbers, d = 1.

Cyclotomic eld

Q(n), n = exp (2i / n)

is a number eld obtained from Q by adjoining a primitive nth root of unity n. This eld containsall complex nth roots of unity and its dimension over Q is equal to (n), where is the Euler totientfunction.

The real numbers,R, and the complex numbers,C, are elds which have innite dimension asQ-vector spaces,hence, they are not number elds. This follows from the uncountability of R and C as sets, whereas everynumber eld is necessarily countable.

The set Q2 of ordered pairs of rational numbers, with the entrywise addition and multiplication is a two-dimensional commutative algebra over Q. However, it is not a eld, since it has zero divisors:

(1, 0) (0, 1) = (1 0, 0 1) = (0, 0).

10 CHAPTER 5. ALGEBRAIC NUMBER FIELD

5.3 Algebraicity and ring of integersGenerally, in abstract algebra, a eld extension F / E is algebraic if every element f of the bigger eld F is the zeroof a polynomial with coecients e0, ..., em in E:

p(f) = emfm + emfm1 + ... + e1f + e0 = 0.

It is a fact that every eld extension of nite degree is algebraic (proof: for x in F simply consider 1, x, x2, x3, ..., weget a linear dependence, i.e. a polynomial that x is a root of!) because of the nite degree. In particular this appliesto algebraic number elds, so any element f of an algebraic number eld F can be written as a zero of a polynomialwith rational coecients. Therefore, elements of F are also referred to as algebraic numbers. Given a polynomialp such that p(f) = 0, it can be arranged such that the leading coecient em is one, by dividing all coecients byit, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rationalcoecients. If, however, its coecients are actually all integers, f is called an algebraic integer. Any (usual) integerz Z is an algebraic integer, as it is the zero of the linear monic polynomial:

p(t) = t z.

It can be shown that any algebraic integer that is also a rational number must actually be an integer, whence thename algebraic integer. Again using abstract algebra, specically the notion of a nitely generated module, it canbe shown that the sum and the product of any two algebraic integers is still an algebraic integer, it follows that thealgebraic integers in F form a ring denoted OF called the ring of integers of F. It is a subring of (that is, a ringcontained in) F. A eld contains no zero divisors and this property is inherited by any subring. Therefore, the ringof integers of F is an integral domain. The eld F is the eld of fractions of the integral domain OF. This way onecan get back and forth between the algebraic number eld F and its ring of integers OF. Rings of algebraic integershave three distinctive properties: rstly, OF is an integral domain that is integrally closed in its eld of fractions F.Secondly, OF is a Noetherian ring. Finally, every nonzero prime ideal of OF is maximal or, equivalently, the Krulldimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (orDedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.

5.3.1 Unique factorization and class numberFor general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product ofprime ideals. However, unlike Z as the ring of integers of Q, the ring of integers of a proper extension of Q neednot admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. Thishappens already for quadratic integers, for example in OQ = Z[5], the uniqueness of the factorization fails:

6 = 2 3 = (1 + 5) (1 5).

Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors donot just dier by a unit in OQ. Euclidean domains are unique factorization domains; for example Z[i], the ringof Gaussian integers, and Z[], the ring of Eisenstein integers, where is a third root of unity (unequal to 1), havethis property.[1]

5.3.2 -functions, L-functions and class number formulaThe failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of theso-called ideal class group. This group is always nite. The ring of integers OF possesses unique factorization if andonly if it is a principal ring or, equivalently, if F has class number 1. Given a number eld, the class number is oftendicult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginaryquadratic number elds (i.e., Q(d), d 1) with prescribed class number. The class number formula relates h toother fundamental invariants of F. It involves the Dedekind zeta function F(s), a function in a complex variable s,dened by

F (s) :=Yp

1

1N(p)s

5.4. BASES FOR NUMBER FIELDS 11

(The product is over all prime ideals of OF, N(p) denotes the norm of the prime ideal or, equivalently, the (-nite) number of elements in the residue eld OF /p . The innite product converges only for Re(s) > 1, in generalanalytic continuation and the functional equation for the zeta-function are needed to dene the function for all s). TheDedekind zeta-function generalizes the Riemann zeta-function in that Q(s) = (s).The class number formula states that F(s) has a simple pole at s = 1 and at this point (its meromorphic continuationto the whole complex plane) the residue is given by

2r1 (2)r2 h Regw pjDj :

Here r1 and r2 classically denote the number of real embeddings and pairs of complex embeddings of F, respectively.Moreover, Reg is the regulator of F, w the number of roots of unity in F and D is the discriminant of F.Dirichlet L-functions L(, s) are a more rened variant of (s). Both types of functions encode the arithmetic behaviorof Q and F, respectively. For example, Dirichlets theorem asserts that in any arithmetic progression

a, a + m, a + 2m, ...

with coprime a andm, there are innitely many prime numbers. This theorem is implied by the fact that the DirichletL-function is nonzero at s = 1. Using much more advanced techniques including algebraic K-theory and Tamagawameasures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture),of values of more general L-functions.[2]

5.4 Bases for number elds

5.4.1 Integral basisAn integral basis for a number eld F of degree n is a set

B = {b1, , bn}

of n algebraic integers in F such that every element of the ring of integers OF of F can be written uniquely as aZ-linear combination of elements of B; that is, for any x in OF we have

x = m1b1 + + mnbn,

where the mi are (ordinary) integers. It is then also the case that any element of F can be written uniquely as

m1b1 + + mnbn,

where now the mi are rational numbers. The algebraic integers of F are then precisely those elements of F where themi are all integers.Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis,and it is now standard for computer algebra systems to have built-in programs to do this.

5.4.2 Power basisLet F be a number eld of degree n. Among all possible bases of F (seen as a Q-vector space), there are particularones known as power bases, that are bases of the form

Bx = {1, x, x2, ..., xn1}

for some element x F. By the primitive element theorem, there exists such an x, called a primitive element. If x canbe chosen in OF and such that Bx is a basis of OF as a free Z-module, then Bx is called a power integral basis, and theeld F is called a monogenic eld. An example of a number eld that is not monogenic was rst given by Dedekind.His example is the eld obtained by adjoining a root of the polynomial x3 x2 2x 8.[3]


5.5 Regular representation, trace and determinant

Using the multiplication in F, the elements of the eld F may be represented by n-by-n matrices

A = A(x)=(aij) i, j n,

by requiring

xei =

nXj=1

aijej ; aij 2 Q:

Here e1, ..., en is a xed basis for F, viewed as a Q-vector space. The rational numbers aij are uniquely determinedby x and the choice of a basis since any element of F can be uniquely represented as a linear combination of the basiselements. This way of associating a matrix to any element of the eld F is called the regular representation. Thesquare matrix A represents the eect of multiplication by x in the given basis. It follows that if the element y of F isrepresented by a matrix B, then the product xy is represented by the matrix product BA. Invariants of matrices, suchas the trace, determinant, and characteristic polynomial, depend solely on the eld element x and not on the basis. Inparticular, the trace of the matrix A(x) is called the trace of the eld element x and denoted Tr(x), and the determinantis called the norm of x and denoted N(x).By denition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a linearfunction of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(x) = Tr(x), and the norm is amultiplicative homogeneousfunction of degree n: N(xy) = N(x) N(y), N(x) = n N(x). Here is a rational number, and x, y are any two elementsof F.The trace form derives is a bilinear form dened by means of the trace, as Tr(x y). The integral trace form, an integer-valued symmetric matrix is dened as t = Tr(bb), where b1, ..., b is an integral basis for F. The discriminant of Fis dened as det(t). It is an integer, and is an invariant property of the eld F, not depending on the choice of integralbasis.The matrix associated to an element x of F can also be used to give other, equivalent descriptions of algebraic integers.An element x of F is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated tox is a monic polynomial with integer coecients. Suppose that the matrix A that represents an element x has integerentries in some basis e. By the CayleyHamilton theorem, pA(A) = 0, and it follows that pA(x) = 0, so that x is analgebraic integer. Conversely, if x is an element of F which is a root of a monic polynomial with integer coecientsthen the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrixin a suitable basis of F. Note that the property of being an algebraic integer is dened in a way that is independent ofa choice of a basis in F.

5.5.1 Example

Consider F = Q(x), where x satises x3 11x2 + x + 1 = 0. Then an integral basis is [1, x, 1/2(x2 + 1)], and thecorresponding integral trace form is

24 3 11 6111 119 65361 653 3589

35:The 3 in the upper left hand corner of this matrix is the trace of the matrix of the map dened by the rst basiselement (1) in the regular representation of F on F. This basis element induces the identity map on the 3-dimensionalvector space, F. The trace of the matrix of the identity map on a 3-dimensional vector space is 3.The determinant of this is 1304 = 23163, the eld discriminant; in comparison the root discriminant, or discriminantof the polynomial, is 5216 = 25163.

5.6. PLACES 13

5.6 PlacesMathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number.[4][5] Thissituation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all ofthe various possible embeddings of a number eld F into its various topological completions at once.A place of a number eld F is an equivalence class of absolute values on F. Essentially, an absolute value is a notion tomeasure the size of elements f of F. Two such absolute values are considered equivalent if they give rise to the samenotion of smallness (or proximity). In general, they fall into three regimes. Firstly (and mostly irrelevant), the trivialabsolute value | |0, which takes the value 1 on all non-zero f in F. The second and third classes are Archimedeanplaces and non-Archimedean (or ultrametric) places. The completion of F with respect to a place is given in bothcases by taking Cauchy sequences in F and dividing out null sequences, that is, sequences (xn)n N such that |xn|tends to zero when n tends to innity. This can be shown to be a eld again, the so-called completion of F at thegiven place.For F = Q, the following non-trivial norms occur (Ostrowskis theorem): the (usual) absolute value, which givesrise to the complete topological eld of the real numbers R. On the other hand, for any prime number p, the p-adicabsolute values is dened by

|q|p = pn, where q = pn a/b and a and b are integers not divisible by p.

In contrast to the usual absolute value, the p-adic norm gets smaller when q is multiplied by p, leading to quite dierentbehavior of Qp vis--vis R.

5.6.1 Archimedean places[6][7]

For some of the details take a look at,[8] Chapter 11 C p. 108. Note in particular the standard notation r1 and r2 forthe number of real and complex embeddings, respectively (see below).Calculating the archimedean places of F is done as follows: let x be a primitive element of F, with minimal polynomial(over Q) f. Over R, f will generally no longer be irreducible, but its irreducible (real) factors are either of degreeone or two. Since there are no repeated roots, there are no repeated factors. The roots r of factors of degree one arenecessarily real, and replacing x by r gives an embedding of F into R; the number of such embeddings is equal to thenumber of real roots of f. Restricting the standard absolute value on R to F gives an archimedean absolute value onF; such an absolute value is also referred to as a real place of F. On the other hand, the roots of factors of degree twoare pairs of conjugate complex numbers, which allows for two conjugate embeddings into C. Either one of this pairof embeddings can be used to dene an absolute value on F, which is the same for both embeddings since they areconjugate. This absolute value is called a complex place of F.If all roots of f above are real (respectively, complex) or, equivalently, any possible embedding F C is actuallyforced to be inside R (resp. C), F is called totally real (resp. totally complex).

5.6.2 Nonarchimedean or ultrametric places

To nd the nonarchimedean places, let again f and x be as above. In Qp, f splits in factors of various degrees, noneof which are repeated, and the degrees of which add up to n, the degree of f. For each of these p-adically irreduciblefactors t, we may suppose that x satises t and obtain an embedding of F into an algebraic extension of nite degreeover Q. Such a local eld behaves in many ways like a number eld, and the p-adic numbers may similarly play therole of the rationals; in particular, we can dene the norm and trace in exactly the same way, now giving functionsmapping toQp. By using this p-adic norm mapNt for the place t, we may dene an absolute value corresponding to agiven p-adically irreducible factor t of degree m by ||t = |Nt()|p1/m. Such an absolute value is called an ultrametric,non-Archimedean or p-adic place of F.For any ultrametric place v we have that |x|v 1 for any x in OF, since the minimal polynomial for x has integerfactors, and hence its p-adic factorization has factors in Zp. Consequently, the norm term (constant term) for eachfactor is a p-adic integer, and one of these is the integer used for dening the absolute value for v.


5.6.3 Prime ideals in OFFor an ultrametric place v, the subset of OF dened by |x|v < 1 is an ideal P of OF. This relies on the ultrametricityof v: given x and y in P, then

|x + y|v max (|x|v, |y|v) < 1.

Actually, P is even a prime ideal.Conversely, given a prime ideal P of OF, a discrete valuation can be dened by setting vP(x) = n where n is thebiggest integer such that x Pn, the n-fold power of the ideal. This valuation can be turned into an ultrametric place.Under this correspondence, (equivalence classes) of ultrametric places of F correspond to prime ideals of OF. ForF = Q, this gives back Ostrowskis theorem: any prime ideal in Z (which is necessarily by a single prime number)corresponds to an non-archimedean place and vice versa. However, for more general number elds, the situationbecomes more involved, as will be explained below.Yet another, equivalent way of describing ultrametric places is by means of localizations of OF. Given an ultrametricplace v on a number eld F, the corresponding localization is the subring T of F of all elements x such that | x |v 1.By the ultrametric property T is a ring. Moreover, it contains OF. For every element x of F, at least one of x or x1is contained in T. Actually, since F/T can be shown to be isomorphic to the integers, T is a discrete valuation ring,in particular a local ring. Actually, T is just the localization of OF at the prime ideal P. Conversely, P is the maximalideal of T.Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations ona number eld.

5.7 Ramication

Schematic depiction of ramication: the bers of almost all points in Y below consist of three points, except for two points in Ymarked with dots, where the bers consist of one and two points (marked in black), respectively. The map f is said to be ramied inthese points of Y.

Ramication, generally speaking, describes a geometric phenomenon that can occur with nite-to-one maps (that is,maps f: X Y such that the preimages of all points y in Y consist only of nitely many points): the cardinality ofthe bers f1(y) will generally have the same number of points, but it occurs that, in special points y, this numberdrops. For example, the map

C C, z zn

has n points in each ber over t, namely the n (complex) roots of t, except in t = 0, where the ber consists of only oneelement, z = 0. One says that the map is ramied in zero. This is an example of a branched covering of Riemannsurfaces. This intuition also serves to dene ramication in algebraic number theory. Given a (necessarily nite)extension of number elds F / E, a prime ideal p of OE generates the ideal pOF of OF. This ideal may or may not bea prime ideal, but, according to the LaskerNoether theorem (see above), always is given by

5.8. GALOIS GROUPS AND GALOIS COHOMOLOGY 15

pOF = q1e1 q2e2 ... qmem

with uniquely determined prime ideals qi of OF and numbers (called ramication indices) ei. Whenever one rami-cation index is bigger than one, the prime p is said to ramify in F.The connection between this denition and the geometric situation is delivered by the map of spectra of rings SpecOFSpecOE. In fact, unramied morphisms of schemes in algebraic geometry are a direct generalization of unramiedextensions of number elds.Ramication is a purely local property, i.e., depends only on the completions around the primes p and qi. The inertiagroup measures the dierence between the local Galois groups at some place and the Galois groups of the involvednite residue elds.

5.7.1 An exampleThe following example illustrates the notions introduced above. In order to compute the ramication index of Q(x),where

f(x) = x3 x 1 = 0,

at 23, it suces to consider the eld extensionQ23(x) /Q23. Up to 529 = 232 (i.e., modulo 529) f can be factored as

f(x) = (x + 181)(x2 181x 38) = gh.

Substituting x = y + 10 in the rst factor gmodulo 529 yields y + 191, so the valuation | y |g for y given by g is | 191|23 = 1. On the other hand the same substitution in h yields y2 161y 161 modulo 529. Since 161 = 7 23,

|y|h = 16123 = 1 / 23.

Since possible values for the absolute value of the place dened by the factor h are not conned to integer powers of23, but instead are integer powers of the square root of 23, the ramication index of the eld extension at 23 is two.The valuations of any element of F can be computed in this way using resultants. If, for example y = x2 x 1, usingthe resultant to eliminate x between this relationship and f = x3 x 1 = 0 gives y3 5y2 + 4y 1 = 0. If insteadwe eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y,and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g andh (which are both 1 in this instance.)

5.7.2 Dedekind discriminant theoremMuch of the signicance of the discriminant lies in the fact that ramied ultrametric places are all places obtainedfrom factorizations inQpwhere p divides the discriminant. This is even true of the polynomial discriminant; howeverthe converse is also true, that if a prime p divides the discriminant, then there is a p-place which ramies. For thisconverse the eld discriminant is needed. This is the Dedekind discriminant theorem. In the example above, thediscriminant of the number eld Q(x) with x3 x 1 = 0 is 23, and as we have seen the 23-adic place ramies.The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramied place comes fromthe absolute value on the complex embedding of F.

5.8 Galois groups and Galois cohomologyGenerally in abstract algebra, eld extensionsF /E can be studied by examining theGalois groupGal(F /E), consistingof eld automorphisms of F leaving E elementwise xed. As an example, the Galois group Gal (Q(n) / Q) of thecyclotomic eld extension of degree n (see above) is given by (Z/nZ), the group of invertible elements in Z/nZ. Thisis the rst stepstone into Iwasawa theory.In order to include all possible extensions having certain properties, the Galois group concept is commonly appliedto the (innite) eld extension F / F of the algebraic closure, leading to the absolute Galois group G := Gal(F / F)


or just Gal(F), and to the extension F / Q. The fundamental theorem of Galois theory links elds in between F andits algebraic closure and closed subgroups of Gal (F). For example, the abelianization (the biggest abelian quotient)Gab of G corresponds to a eld referred to as the maximal abelian extension Fab (called so since any further extensionis not abelian, i.e., does not have an abelian Galois group). By the KroneckerWeber theorem, the maximal abelianextension of Q is the extension generated by all roots of unity. For more general number elds, class eld theory,specically the Artin reciprocity law gives an answer by describing Gab in terms of the idele class group. Also notableis the Hilbert class eld, the maximal abelian unramied eld extension of F. It can be shown to be nite over F, itsGalois group over F is isomorphic to the class group of F, in particular its degree equals the class number h of F (seeabove).In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is thenalso referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal(F), alsoknown as Galois cohomology, which in the rst place measures the failure of exactness of taking Gal(F)-invariants,but oers deeper insights (and questions) as well. For example, the Galois group G of a eld extension L / F actson L, the nonzero elements of L. This Galois module plays a signicant role in many arithmetic dualities, such asPoitou-Tate duality. The Brauer group of F, originally conceived to classify division algebras over F, can be recast asa cohomology group, namely H2(Gal (F), F).

5.9 Local-global principle

Generally speaking, the term local to global refers to the idea that a global problem is rst done at a local level,which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be puttogether to get back to some global statement. For example, the notion of sheaves reies that idea in topology andgeometry.

5.9.1 Local and global elds

Number elds share a great deal of similarity with another class of elds much used in algebraic geometry known asfunction elds of algebraic curves over nite elds. An example is Fp(T). They are similar in many respects, for ex-ample in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient elds of whichis the function eld in question) of curves. Therefore, both types of eld are called global elds. In accordance withthe philosophy laid out above, they can be studied at a local level rst, that is to say, by looking at the correspondinglocal elds. For number elds F, the local elds are the completions of F at all places, including the archimedeanones (see local analysis). For function elds, the local elds are completions of the local rings at all points of thecurve for function elds.Many results valid for function elds also hold, at least if reformulated properly, for number elds. However, thestudy of number elds often poses diculties and phenomena not encountered in function elds. For example, infunction elds, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function eldsoften serves as a source of intuition what should be expected in the number eld case.

5.9.2 Hasse principle

A prototypical question, posed at a global level, is whether some polynomial equation has a solution in F. If thisis the case, this solution is also a solution in all completions. The local-global principle or Hasse principle assertsthat for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solutioncan be done on all the completions of F, which is often easier, since analytic methods (classical analytic tools suchas intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can beused. This implication does not hold, however, for more general types of equations. However, the idea of passingfrom local data to global ones proves fruitful in class eld theory, for example, where local class eld theory is usedto obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completionsF can be explicitly determined, whereas the Galois groups of global elds, even of Q are far less understood.

5.10. SEE ALSO 17

5.9.3 Adeles and idelesIn order to assemble local data pertaining to all local elds attached to F, the adele ring is set up. A multiplicativevariant is referred to as ideles.

5.10 See also Dirichlets unit theorem, S-unit Kummer extension Minkowskis theorem, Geometry of numbers Chebotarevs density theorem Ray class group Decomposition group Genus eld

5.11 Notes[1] Ireland, Kenneth; Rosen, Michael (1998), A Classical Introduction to Modern Number Theory, Berlin, New York: Springer-

Verlag, ISBN 978-0-387-97329-6, Ch. 1.4

[2] Bloch, Spencer; Kato, Kazuya (1990), "L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol.I, Progr. Math. 86, Boston, MA: Birkhuser Boston, pp. 333400, MR 1086888

[3] Narkiewicz 2004, 2.2.6

[4] Kleiner, Israel (1999), Field theory: from equations to axiomatization. I, The American Mathematical Monthly 106 (7):677684, doi:10.2307/2589500, MR 1720431, To Dedekind, then, elds were subsets of the complex numbers.

[5] Mac Lane, Saunders (1981), Mathematical models: a sketch for the philosophy of mathematics, The American Mathe-matical Monthly 88 (7): 462472, doi:10.2307/2321751, MR 628015, Empiricism sprang from the 19th-century view ofmathematics as almost coterminal with theoretical physics.

[6] Cohn

[7] Conrad

[8] Cohn

5.12 References Cohn, Harvey (1988), A Classical Invitation to Algebraic Numbers and Class Fields, Universitext, New York:Springer-Verlag

Conrad, Keith http://www.math.uconn.edu/~{}kconrad/blurbs/gradnumthy/unittheorem.pdf Janusz, Gerald J. (1996), Algebraic Number Fields (2nd ed.), Providence, R.I.: American Mathematical Soci-ety, ISBN 978-0-8218-0429-2

Helmut Hasse, Number Theory, Springer Classics in Mathematics Series (2002) Serge Lang, Algebraic Number Theory, second edition, Springer, 2000 Richard A. Mollin, Algebraic Number Theory, CRC, 1999 Ram Murty, Problems in Algebraic Number Theory, Second Edition, Springer, 2005


Narkiewicz, Wadysaw (2004), Elementary and analytic theory of algebraic numbers, Springer Monographsin Mathematics (3 ed.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267

Neukirch, Jrgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322,Berlin, New York: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021

Neukirch, Jrgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehrender MathematischenWissenschaften 323, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66671-4, MR1737196, Zbl 1136.11001

Andr Weil, Basic Number Theory, third edition, Springer, 1995

Chapter 6

Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is dened to be the smallest number of timesone must use the rings multiplicative identity element (1) in a sum to get the additive identity element (0); the ring issaid to have characteristic zero if this sum never reaches the additive identity.That is, char(R) is the smallest positive number n such that

1 + + 1| {z }nsummands

= 0

if such a number n exists, and 0 otherwise.The characteristic may also be taken to be the exponent of the rings additive group, that is, the smallest positive nsuch that

a+ + a| {z }nsummands

= 0

for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicativeidentity element in their requirements for a ring (see ring), and this denition is suitable for that convention; otherwisethe two denitions are equivalent due to the distributive law in rings.

6.1 Other equivalent characterizations The characteristic is the natural number n such that nZ is the kernel of a ring homomorphism from Z to R; The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ,which would be the image of that homomorphism.

When the non-negative integers {0, 1, 2, 3, . . . } are partially ordered by divisibility, then 1 is the smallestand 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n 1 = 0. If nothingsmaller (in this ordering) than 0 will suce, then the characteristic is 0. This is the right partial orderingbecause of such facts as that char A B is the least common multiple of char A and char B, and that no ringhomomorphism : A B exists unless char B divides char A.

The characteristic of a ring R is n {0, 1, 2, 3, . . . } precisely if the statement ka = 0 for all a R implies nis a divisor of k.

The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring ofintegers to any ring; in the language of category theory, Z is an initial object of the category of rings. Again thisfollows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms).

19

20 CHAPTER 6. CHARACTERISTIC (ALGEBRA)

6.2 Case of ringsIf R and S are rings and there exists a ring homomorphism R S, then the characteristic of S divides the characteristicof R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring withcharacteristic 1 is the trivial ring which has only a single element 0 = 1. If a non-trivial ring R does not have any zerodivisors, then its characteristic is either 0 or prime. In particular, this applies to all elds, to all integral domains, andto all division rings. Any ring of characteristic 0 is innite.The ring Z/nZ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same char-acteristic. For instance, if q(X) is a prime polynomial with coecients in the eld Z/pZ where p is prime, then thefactor ring (Z/pZ)[X]/(q(X)) is a eld of characteristic p. Since the complex numbers contain the rationals, theircharacteristic is 0.A Z/nZ-algebra is equivalently a ring whose characteristic divides n.If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R the "freshmans dream" holds for power p.The map

f(x) = xp

then denes a ring homomorphism

R R.

It is called the Frobenius homomorphism. If R is an integral domain it is injective.

6.3 Case of eldsAs mentioned above, the characteristic of any eld is either 0 or a prime number. A eld of non-zero characteristicis called a eld of nite characteristic or a eld of positive characteristic.For any eldF, there is aminimal subeld, namely the prime eld, the smallest subeld containing 1F. It is isomorphiceither to the rational number eld Q, or a nite eld of prime order, Fp; the structure of the prime eld and thecharacteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practicalpurposes they resemble subelds of the complex numbers (unless they have very large cardinality, that is; in fact, anyeld of characteristic zero and cardinality at most continuum is isomorphic to a subeld of complex numbers).[1] Thep-adic elds or any nite extension of them are characteristic zero elds, much applied in number theory, that areconstructed from rings of characteristic pk, as k .For any ordered eld, as the eld of rational numbers Q or the eld of real numbers R, the characteristic is 0. Thus,number elds and the eld of complex numbers C are of characteristic zero. Actually, every eld of characteristiczero is the quotient eld of a ring Q[X]/P where X is a set of variables and P a set of polynomials in Q[X]. Thenite eld GF(pn) has characteristic p. There exist innite elds of prime characteristic. For example, the eld ofall rational functions over Z/pZ, the algebraic closure of Z/pZ or the eld of formal Laurent series Z/pZ((T)). Thecharacteristic exponent is dened similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has thesame value as the characteristic.[2]

The size of any nite ring of prime characteristic p is a power of p. Since in that case it must contain Z/pZ it mustalso be a vector space over that eld and from linear algebra we know that the sizes of nite vector spaces over niteelds are a power of the size of the eld. This also shows that the size of any nite vector space is a prime power. (Itis a vector space over a nite eld, which we have shown to be of size pn. So its size is (pn)m = pnm.)

6.4 References[1] Enderton, Herbert B. (2001),AMathematical Introduction to Logic (2nd ed.), Academic Press, p. 158, ISBN9780080496467.

Enderton states this result explicitly only for algebraically closed elds, but also describes a decomposition of any eld asan algebraic extension of a transcendental extension of its prime eld, from which the result follows immediately.

6.4. REFERENCES 21

[2] Field Characteristic Exponent. Wolfram Mathworld. Wolfram Research. Retrieved May 27, 2015.

Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.

Chapter 7

Countable set

Countable redirects here. For the linguistic concept, see Count noun.Not to be confused with (recursively) enumerable sets.

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the setof natural numbers. A countable set is either a nite set or a countably innite set. Whether nite or innite, theelements of a countable set can always be counted one at a time and, although the counting may never nish, everyelement of the set is associated with a natural number.Some authors use countable set to mean innitely countable alone.[1] To avoid this ambiguity, the term at mostcountable may be used when nite sets are included and countably innite, enumerable, or denumerable[2] oth-erwise.The term countable set was originated by Georg Cantor who contrasted sets which are countable with those which areuncountable (a.k.a. nonenumerable and nondenumerable[3]). Today, countable sets are researched by a branch ofmathematics called discrete mathematics.

7.1 DenitionA set S is called countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}.[4]

If such an f can be found which is also surjective (and therefore bijective), then S is called countably innite.In other words, a set is called countably innite if it has one-to-one correspondence with the natural number set, N.As noted above, this terminology is not universal: Some authors use countable to mean what is here called countablyinnite, and to not include nite sets.For alternative (equivalent) formulations of the denition in terms of a bijective function or a surjective function, seethe section Formal denition and properties below.

7.2 HistoryIn the western world, dierent innities were rst classied by Georg Cantor around 1874.[5]

7.3 IntroductionA set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements;for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}. This is only eective for smallsets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element,sometimes an ellipsis ("...) is used, if the writer believes that the reader can easily guess what is missing; for example,

22

7.4. FORMAL DEFINITION AND PROPERTIES 23

{1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possibleto list all the elements, because the set is nite.Some sets are innite; these sets have more than n elements for any integer n. For example, the set of natural numbers,denotable by {0, 1, 2, 3, 4, 5, ...}, has innitely many elements, and we cannot use any normal number to give itssize. Nonetheless, it turns out that innite sets do have a well-dened notion of size (or more properly, of cardinality,which is the technical term for the number of elements in a set), and not all innite sets have the same cardinality.

YX123

x

246

2x. .

. .

Bijective mapping from integer to even numbers

To understand what this means, we rst examine what it does not mean. For example, there are innitely many oddintegers, innitely many even integers, and (hence) innitely many integers overall. However, it turns out that thenumber of even integers, which is the same as the number of odd integers, is also the same as the number of integersoverall. This is because we arrange things such that for every integer, there is a distinct even integer: ... 24,12, 00, 12, 24, ...; or, more generally, n2n, see picture. What we have done here is arranged the integersand the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two setssuch that each element of each set corresponds to a single element in the other set.However, not all innite sets have the same cardinality. For example, Georg Cantor (who introduced this concept)demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.A set is countable if: (1) it is nite, or (2) it has the same cardinality (size) as the set of natural numbers. Equivalently, aset is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable.

7.4 Formal denition and propertiesBy denition a set S is countable if there exists an injective function f : S N from S to the natural numbers N ={0, 1, 2, 3, ...}.

24 CHAPTER 7. COUNTABLE SET

It might seem natural to divide the sets into dierent classes: put all the sets containing one element together; all thesets containing two elements together; ...; nally, put together all innite sets and consider them as having the samesize. This view is not tenable, however, under the natural denition of size.To elaborate this we need the concept of a bijection. Although a bijection seems a more advanced concept than anumber, the usual development of mathematics in terms of set theory denes functions before numbers, as they arebased on much simpler sets. This is where the concept of a bijection comes in: dene the correspondence

a 1, b 2, c 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this denes abijection.We now generalize this situation and dene two sets to be of the same size if (and only if) there is a bijection betweenthem. For all nite sets this gives us the usual denition of the same size. What does it tell us about the size ofinnite sets?Consider the sets A = {1, 2, 3, ... }, the set of positive integers and B = {2, 4, 6, ... }, the set of even positive integers.We claim that, under our denition, these sets have the same size, and that therefore B is countably innite. Recallthat to prove this we need to exhibit a bijection between them. But this is easy, using n 2n, so that

1 2, 2 4, 3 6, 4 8, ....

As in the earlier example, every element of A has been paired o with precisely one element of B, and vice versa.Hence they have the same size. This gives an example of a set which is of the same size as one of its proper subsets,a situation which is impossible for nite sets.Likewise, the set of all ordered pairs of natural numbers is countably innite, as can be seen by following a path likethe one in the picture:The resulting mapping is like this:

0 (0,0), 1 (1,0), 2 (0,1), 3 (2,0), 4 (1,1), 5 (0,2), 6 (3,0) ....

It is evident that this mapping will cover all such ordered pairs.Interestingly: if you treat each pair as being the numerator and denominator of a vulgar fraction, then for everypositive fraction, we can come up with a distinct number corresponding to it. This representation includes also thenatural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as manypositive rational numbers as there are positive integers. This is true also for all rational numbers, as can be seen below(a more complex presentation is needed to deal with negative numbers).Theorem: The Cartesian product of nitely many countable sets is countable.This form of triangular mapping recursively generalizes to vectors of nitely many natural numbers by repeatedlymapping the rst two elements to a natural number. For example, (0,2,3) maps to (5,3) which maps to 39.Sometimes more than onemapping is useful. This is where youmap the set which you want to show countably innite,onto another set; and then map this other set to the natural numbers. For example, the positive rational numbers caneasily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (p, q).What about innite subsets of countably innite sets? Do these have fewer elements than N?Theorem: Every subset of a countable set is countable. In particular, every innite subset of a countably innite setis countably innite.For example, the set of prime numbers is countable, by mapping the n-th prime number to n:

2 maps to 1 3 maps to 2 5 maps to 3 7 maps to 4


1

2

3

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The Cantor pairing function assigns one natural number to each pair of natural numbers

11 maps to 5 13 maps to 6 17 maps to 7 19 maps to 8 23 maps to 9 ...

What about sets being larger than N? An obvious place to look would be Q, the set of all rational numbers, whichintuitively may seem much bigger than N. But looks can be deceiving, for we assert:Theorem: Q (the set of all rational numbers) is countable.Q can be dened as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto thesubset of ordered triples of natural numbers (a, b, c) such that a 0, b > 0, a and b are coprime, and c {0, 1} suchthat c = 0 if a/b 0 and c = 1 otherwise.

0 maps to (0,1,0)


1 maps to (1,1,0) 1 maps to (1,1,1) 1/2 maps to (1,2,0) 1/2 maps to (1,2,1) 2 maps to (2,1,0) 2 maps to (2,1,1) 1/3 maps to (1,3,0) 1/3 maps to (1,3,1) 3 maps to (3,1,0) 3 maps to (3,1,1) 1/4 maps to (1,4,0) 1/4 maps to (1,4,1) 2/3 maps to (2,3,0) 2/3 maps to (2,3,1) 3/2 maps to (3,2,0) 3/2 maps to (3,2,1) 4 maps to (4,1,0) 4 maps to (4,1,1) ...

By a similar development, the set of algebraic numbers is countable, and so is the set of denable numbers.Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable.For example, given countable sets a, b, c ...Using a variant of the triangular enumeration we saw above:

a0 maps to 0 a1 maps to 1 b0 maps to 2 a2 maps to 3 b1 maps to 4 c0 maps to 5 a3 maps to 6 b2 maps to 7 c1 maps to 8 d0 maps to 9 a4 maps to 10 ...


Note that this only works if the sets a, b, c,... are disjoint. If not, then the union is even smaller and is therefore alsocountable by a previous theorem.Also note that the axiom of countable choice is needed in order to index all of the sets a, b, c,...Theorem: The set of all nite-length sequences of natural numbers is countable.This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which isa countable set (nite Cartesian product). So we are talking about a countable union of countable sets, which iscountable by the previous theorem.Theorem: The set of all nite subsets of the natural numbers is countable.If you have a nite subset, you can order the elements into a nite sequence. There are only countably many nitesequences, so also there are only countably many nite subsets.The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. A proofof this result can be found in Langs text.[2]

Theorem: Let S be a set. The following statements are equivalent:

1. S is countable, i.e. there exists an injective function f : S N.

2. Either S is empty or there exists a surjective function g : N S.

3. Either S is nite or there exists a bijection h : N S.

Several standard properties follow easily from this theorem. We present them here tersely. For a gentler presentationsee the sections above. Observe that N in the theorem can be replaced with any countably innite set. In particularwe have the following Corollary.Corollary: Let S and T be sets.

1. If the function f : S T is injective and T is countable then S is countable.

2. If the function g : S T is surjective and S is countable then T is countable.

Proof: For (1) observe that if T is countable there is an injective function h : T N. Then if f : S T is injectivethe composition h o f : S N is injective, so S is countable.For (2) observe that if S is countable there is a surjective function h : N S. Then if g : S T is surjective thecomposition g o h : N T is surjective, so T is countable.Proposition: Any subset of a countable set is countable.Proof: The restriction of an injective function to a subset of its domain is still injective.Proposition: The Cartesian product of two countable sets A and B is countable.Proof: Note that N N is countable as a consequence of the denition because the function f : N N N givenby f(m, n) = 2m3n is injective. It then follows from the Basic Theorem and the Corollary that the Cartesian productof any two countable sets is countable. This follows because if A and B are countable there are surjections f : N A and g : N B. So

f g : N N A B

is a surjection from the countable set N N to the set A B and the Corollary implies A B is countable. This resultgeneralizes to the Cartesian product of any nite collection of countable sets and the proof follows by induction onthe number of sets in the collection.Proposition: The integers Z are countable and the rational numbers Q are countable.Proof: The integers Z are countable because the function f : Z N given by f(n) = 2n if n is non-negative and f(n)= 3|n| if n is negative is an injective function. The rational numbers Q are countable because the function g : Z N Q given by g(m, n) = m/(n + 1) is a surjection from the countable set Z N to the rationals Q.Proposition: If An is a countable set for each n in N then the union of all An is also countable.


Proof: This is a consequence of the fact that for each n there is a surjective function gn : N An and hence thefunction

G : N N![n2N

An

given by G(n, m) = gn(m) is a surjection. Since N N is countable, the Corollary implies that the union is countable.We are using the axiom of countable choice in this proof in order to pick for each n in N a surjection gn from thenon-empty collection of surjections from N to An.Cantors Theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is nosurjective function from A to P(A). A proof is given in the article Cantors Theorem. As an immediate consequenceof this and the Basic Theorem above we have:Proposition: The set P(N) is not countable; i.e. it is uncountable.For an elaboration of this result see Cantors diagonal argument.The set of real numbers is uncountable (see Cantors rst uncountability proof), and so is the set of all innitesequences of natural numbers. A topological proof for the uncountability of the real numbers is described at niteintersection property.

7.5 Minimal model of set theory is countableIf there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standardmodel (see Constructible universe). The Lwenheim-Skolem theorem can be used to show that this minimal modelis countable. The fact that the notion of uncountability makes sense even in this model, and in particular that thismodel M contains elements which are

subsets of M, hence countable,

but uncountable from the point of view of M,

was seen as paradoxical in the early days of set theory, see Skolems paradox.The minimal standard model includes all the algebraic numbers and all eectively computable transcendental num-bers, as well as many other kinds of numbers.

7.6 Total ordersCountable sets can be totally ordered in various ways, e.g.:

Well orders (see also ordinal number):

The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...) The integers in the order (0, 1, 2, 3, ...; 1, 2, 3, ...)

Other (not well orders):

The usual order of integers (..., 3, 2, 1, 0, 1, 2, 3, ...) The usual order of rational numbers (Cannot be explicitly written as an ordered list!)

Note that in both examples of well orders here, any subset has a least element; and in both examples of non-wellorders, some subsets do not have a least element. This is the key denition that determines whether a total order isalso a well order.

7.7. SEE ALSO 29

7.7 See also Aleph number Counting Hilberts paradox of the Grand Hotel Uncountable set

7.8 Notes[1] For an example of this usage see (Rudin 1976, Chapter 2).

[2] See (Lang 1993, 2 of Chapter I).

[3] See (Apostol 1969, Chapter 13.19).

[4] Since there is an obvious bijection between N and N* = {1, 2, 3, ...}, it makes no dierence whether one considers 0 tobe a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic,which make 0 a natural number.

[5] Stillwell, John C. (2010), Roads to Innity: TheMathematics of Truth and Proof, CRC Press, p. 10, ISBN 9781439865507,Cantors discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before1874, innity was not even considered a legitimate mathematical subject by most people, so the need to distinguish betweencountable and uncountable innities could not have been imagined.

7.9 References Lang, Serge (1993), Real and Functional Analysis, Berlin, New York: Springer-Verlag, ISBN 0-387-94001-4 Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN 0-07-054235-X Apostol, Tom M. (June 1969),Multi-Variable Calculus and Linear Algebra with Applications, Calculus 2 (2nded.), New York: John Wiley + Sons, ISBN 978-0-471-00007-5

7.10 External links Weisstein, Eric W., Countable Set, MathWorld.

Chapter 8

Degree of a eld extension

In mathematics, more specically eld theory, the degree of a eld extension is a rough measure of the size ofthe eld extension. The concept plays an important role in many parts of mathematics, including algebra and numbertheory indeed in any area where elds appear prominently.

8.1 Denition and notation

Suppose that E/F is a eld extension. Then E may be considered as a vector space over F (the eld of scalars). Thedimension of this vector space is called the degree of the eld extension, and it is denoted by [E:F].The degree may be nite or innite, the eld being called a nite extension or innite extension accordingly. Anextension E/F is also someti

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