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Irreducible Polynomials

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Characters are:

Q. Z(2)[x] (mod 2), Z[x], Q, Z(2), Z(3), Z(5), Z(9). all mods

? Decide, giving your reasons, which of the following polynomials is irreducible over . Factorize those polynomials that are reducible into a product of irreducible factors.

a)
b)
c)

? Find, with justification, all monic irreducible cubics in .
? Find, with justification, a polynomial in that is irreducible over but not over .

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Irreducible polynomials are investigated.

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Please see the attached file for the complete solution.

1 Decide, giving your reasons, which of the following polynomials is irreducible over . Factorise those polynomials that are reducible into a product of irreducible factors.

a)
b)
c)

2 Find, with justification, all monic irreducible cubics in .
3 Find, with justification, a polynomial in that is irreducible over but not over .

Characters are: Q, Z(2)[x] (mod 2), Z[x], Q, Z(2), Z(3), Z(5), Z(9). all mods

Solutions:

1. a ) Consider the function , f (x) =

Going by Descartes Rule of Sign Change , since there being no sign change in their co-efficients , we say that ... there is no positive zero's for this polynomial , f (x) .

f (-x) = ... number of sign change = 3

So , possible number of negative roots = 3 or 1

We then use Rational Root Test .. to get the possible rational roots as .

Since there is no positive root , only possible zero = - 1 . Root = x + 1

Now from synthetic division ... we get the reduced equation as :
f (x) =

b ) Consider the function , ...

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