Download - Gallian Ch 17
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Irreducible Polynomial
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Let D be an integral domain.
A polynomial ( )f x from [ ]D x that is neither the zero
polynomial nor a unit in [ ]D x , such that whenever ( )f x
is expressed as a product ( ) ( ) ( )f x g x h x , with ( )g x
and ( )h x from [ ]D x , then ( )g x or ( )h x is a unit in[ ]D x .
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Reducible Polynomial
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Let D be an integral domain.
A nonzero, nonunit polynomial from [ ]D x that is notirreducible.
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Reducibility Test for Degrees 2 and 3
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Let Fbe a field. If ( ) [ ]f x F x and deg ( ) 2f x or 3,
then( )f x
is reducible overF if and only if( )f x
has a
zero in F.
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Content of a polynomial
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Polynomial coefficients are integer.
The greatest common divisor of the polynomialcoefficients.
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Primitive Polynomial
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A polynomial in [ ]Z x with content 1.
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Gausss Lemma
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The product of two primitive polynomials is primitive.
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OverQ Implies OverZ
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Let ( ) [ ]f x Z x . If ( )f x is reducible overQ, then it isreducible overZ.
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Mod p Irreducibility Test
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Let p be a prime and suppose that ( ) [ ]f x Z x with
deg ( ) 1f x . Let
( )f xbe the polynomial in
[ ]pZ x
obtained from ( )f x by reducing all of its coefficients
mod p . If ( )f x is irreducible over pZ and
deg ( ) deg ( )f x f x , then ( )f x is irreducible overQ.
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Eisensteins Criterion
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Let0
( ) ... [ ]nnf x a x a Z x . If there is a prime p
such that p divides every coefficient of ( )f x except for
na and2p does not divide
0a , then ( )f x is irreducible
overQ.
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Irreducibility ofp-th Cyclotomic Polynomial
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For any prime p, the p-th cyclotomic polynomial
1 21( ) ... 11
p p pp
xx x x xx
is irreducible overQ.
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( )p x Is Irreducible if and Only if ( )p x is Maximal
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Let Fbe a field and ( ) [ ]p x F x . Then ( )p x is a
maximal ideal in [ ]F x if and only if ( )p x is irreducible
overF.
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[ ]/ ( )F x p x Is a Field
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Let Fbe a field and ( )p x an irreducible polynomial over
F. Then [ ]/ ( )F x p x is a field.
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( )| ( ) ( )p x a x b x Implies ( )| ( )p x a x or ( )| ( )p x b x
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Let Fbe a field and let ( ), ( ), ( ) [ ]p x a x b x F x . If ( )p x
is irreducible overFand ( )| ( ) ( )p x a x b x , then ( )| ( )p x a x
or ( )| ( )p x b x .
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Unique Factorization in [ ]Z x
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Every polynomial in [ ]Z x that is not the zero polynomial
or a unit in [ ]Z x can be written in the form
1 2 1 2... ( ) ( )... ( )s mb b b p x p x p x , where the ib s are
irreducible polynomials of degree 0, and the ( )ip x s are
irreducible polynomials of positive degree.
Furthermore, if 1 2 1 2... ( ) ( )... ( )s mb b b p x p x p x
1 2 1 2... ( ) ( )... ( )ntc c qc x q x q x , where the ic s are irreducible
polynomials of degree 0, and the ( )iq x s are irreducible
polynomials of positive degree, then ,s t m n , and,
after renumbering the cs and q(x)s, we have i ib c for 1,...,i s and ( ) ( )i ip x q x for 1,...,i m .