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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 .