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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Solution Summary
Irreducible polynomials are investigated.
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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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