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Section Irreducible Polynomials
Worksheet Part 1: Ideals in \(F[x]\) and Irreducibility Basics
These problems are designed to help you understand the basics of irreducibility and working with ideals in
\(F[x]\text{.}\)
1. Find
\(f \in \Z[x]\) with
\(1/2\) and
\(1/3\) as roots
2. Show that
\(I=\{f \in \Z[x] \mid a_n+a_{n-1}+\ldots+a_1+a_0=0\}\) is an ideal of
\(F[x]\) if
\(F\) is a field and find a generator.
3. Find an example of a polynomial
\(f\in\Q[x]\) of degree larger than 3 which is reducible but has no zeros in
\(\Q\text{.}\)
Lemma 192 . Gauss’s Lemma.
The product of two primitive polynomials is primitive.
4. Prove Gauss’s Lemma by using the induced homomorphism
\(\phi:\Z[x]\to \Z_p[x]\) for a prime
\(p\) to show that if
\(p\) divides the content of
\(f(x)g(x)\text{,}\) then
\(p\) divides the content of
\(f(x)\) or
\(g(x)\text{.}\)
Worksheet Part 2: Tests for Irreducibility
These problems are designed to help you practice applying tests for irreducibility of polynomials over
\(\Q\text{.}\)
1. Show that
\(8x^3-6x+1\) is irreducible over
\(\Q\text{.}\) (Using the translation properties from the Daily Prep will be helpful).
2. Show that
\(f=(3/7)x^4-(2/7)x^2+(9/35)x+3/5\) is irreducible over
\(\Q\) by clearing denominators and applying the Mod
\(p\) test.
3. Show that
\(5x^5-6x^4-3x^2+9x-15\) is irreducible over
\(\Q\text{.}\)
4. Use Eisenstein’s Criterion with the prime
\(5\) to construct two different irreducible polynomials of degree
\(6\) over
\(\Q\text{.}\)
