rational numbers and fields. integers – well ordered integral domain can we solve any linear...
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![Page 1: Rational Numbers and Fields. Integers – Well ordered integral domain Can we solve any linear equation over the integers? Example: x + 5 = 7 3x + 5 = 11](https://reader036.vdocuments.net/reader036/viewer/2022081603/56649f295503460f94c43409/html5/thumbnails/1.jpg)
Rational Numbersand
Fields
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Integers – Well ordered integral domain
•Can we solve any linear equation over the integers?
Example: x + 5 = 7 3x + 5 = 11
•What property do the integers lack that we need to be able to solve the equation on the right?
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Field
• A commutative ring F with unity where every nonzero element of F has a multiplicative inverse in F.
• F must also have more than one element. Why?
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Discussion
• Give at least two examples of fields.
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Finite Fields
• Must a field be an infinite set? Let’s explore.
• Is ( Z4 , + ,• ) a field?
+ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
·0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1
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Finite Fields
• Is ( Z5 , + ,• ) a field? + 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3
· 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1
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Finite Fields
• (Z p , + , • ) where p is prime is a field.
Proof: Verify it is a commutative ring with unity (you can do this).
Verify the existence of an inverse.
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Division Algorithm for Integers
• Let a, b Z with b 0, then there exist unique q, r Z such that
a = b•q + r where 0 < r < | b |
• We name these integers the Dividend a, Divisor b, Quotient q, Remainder r
• Example: 25/7 can be expressed
25 = 7 • 3 + 4 where 0 < 4 < 7
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Euclidean Algorithm
• Greatest Common Divisor (g.c.d) can be found by repeated application of the Division Algorithm.
• Example: gcd(630,66) Generalization: gcd(a1, a2 )
630 = 66•9 + 36 a1 = a2 • q1 + a3
66 = 36•1 + 30 a2 = a3 • q2 + a4
36 = 30•1 + 6 a3 = a4 • q3 + a5 30 = 6•5 + 0
an-2 = an-1 • qn-2 + an
an-1 = an • qn-1
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Finite Fields
• The Euclidean Algorithm provides the existence of the inverse in Zp
Proof (completed): We needed ax + p(-q) = 1. Since p is prime then gcd(a, p) = 1.
So by the Euclidean Algorithm
an-2 = an-1 • qn-2 + 1 or an-2 - an-1 • qn-2 = 1
We can back substitute for the a values to get the desired equation. QED
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Field and Integral Domain
• Is a field F always an integral domain?
• Verify this by letting r,s F such that
r • s = 0 and suppose r 0. What do we have to show?
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Rational Numbers – An Extension of the Integers
• Let S = {(a , b) | a , b Z ,b 0 }
• Think of (a , b) as familiar a / b, but symbol a / b has no meaning until there is a field containing a and b.
• Want a / b = a•n / b•n for any n Z, n0. So need (a ,b) (an ,bn)
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Rational Numbers – An Extension of the Integers
• Define equivalence relation (a ,b) (c,d) only if ad = bc.
• Verify this is an equivalence relation.
• Consider Equivalence Classes
[a, b] = {(x ,y) | (x ,y) S and ay = bx}
• Provide an example of an equivalence class
• Let our new field F = { [a, b ] | (a ,b) S}
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Binary Operations on set F = { [a ,b] | (a , b) S }
• Define so they parallel + and • of rational numbers
• Addition: [a ,b] + [c , d] =[ad+bc,bd]
• Multiplication: [a, b] • [c ,d] = [ac,bd]
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• Closure: For all x , y Set, x+y Set and x • y Set.
Well Defined Operation:
If X = X1 and Y = Y1 then X + Y = X1+ Y1
If X = X1 and Y = Y1 then X • Y = X1 • Y1
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(F,+,•) field of Rational NumbersVerify the field propertiesAddition Properties Multiplication PropertiesClosure ClosureIdentity IdentityInverse InverseCommutative CommutativeAssociative Associative
Distributive Property
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Quotient Field
• What is the additive identity?
• What is the additive inverse?
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Quotient Field
• What is the Multiplicative Identity?
• What is the Multiplicative Inverse?
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Question
• In extending D to F, why is it necessary that D be an integral domain, and not just a commutative ring with unity?
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Rational OrderIs (Q,+,•) an ordered integral domain?
Recall the definition of ordered.
Ordered Integral Domain: Contains a subset D+ with the following properties.
1. If a, b D+ ,then a + b D+ (closure)
2. If a , b D+ , then a • b D+
3. For each a Integral Domain D exactly one of these holds
a = 0, a D+ , -a D+ (Trichotomy)
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• Thank you!