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Chapter 14: Problem 15
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Let \(F\) be a field, and let \(f, g_{1}, \ldots, g_{m} \in F[x]\) be monic nonconstant polynomials. Recall that \(\left(g_{1}, \ldots, g_{m}\right)\) is the squarefree decomposition of \(f\) if \(f=g_{1} g_{2}^{2} \cdots g_{m}^{m}\), each \(g_{i}\) is squarefree, the \(g_{i}\) are pairwise coprime, and \(g_{m} \neq 1\). (i) Prove that there is a unique decomposition \(f=h_{1} \cdots h_{m}\) such that each \(h_{i}\) is monic, nonconstant, and squarefree, \(h_{i} \mid h_{i-1}\) for \(2 \leq i \leq m\), and \(h_{m} \neq 1\). (ii) Give both decompositions for \(f=x^{4}(x+1)^{3}\). (iii) Express the \(h_{i}\) in terms of the \(g_{i}\) and vice versa, and show that both conversions can be computed in time \(O(\mathrm{M}(n))\) if \(n=\operatorname{deg} f\).
Let \(q\) be a prime power, \(t \in \mathbb{N}\) a prime divisor of \(q-1\), and \(a \in \mathbb{F}_{q}^{\times}\). (i) Show that the polynomial \(x^{t}-a \in \mathbb{F}_{q}[x]\) splits into linear factors if \(a\) is a \(t\) th power (ii) Show that \(x^{t}-a\) is irreducible if \(a\) is not a \(t\) th power Hint: Use (i) for the splitting field of \(x^{t}-a\) and consider the constant coefficient of a hypothetical factor \(f \in \mathbb{F}_{q}[x]\) of \(x^{t}-a\). (iii) Derive a formula for the probability that a random binomial \(x^{t}-a\) (that is, for random \(a \in \mathbb{F}_{q}^{\times}\)) is irreducible, and compare it to the probability that a random polynomial of degree \(t\) in \(\mathbb{F}_{q}[x]\) is irreducible.
Test the following polynomials for multiple factors in \(\mathbb{Q}[x]\). (i) \(x^{3}-3 x^{2}+4\) (ii) \(x^{3}-2 x^{2}-x+2\).
Let \(q\) be a prime power and \(f \in \mathbb{F}_{q}[x]\) squarefree of degree
\(n\).
(i) Prove that for \(1 \leq a \leq b \leq n\), the polynomial
$$
\operatorname{gcd}\left(\prod_{a \leq d
Compute the squarefree decomposition of the following polynomials in \(\mathbb{Q}[x]\) and in \(\mathbb{F}_{3}[x]\). (i) \(x^{6}-x^{5}-4 x^{4}+2 x^{3}+5 x^{2}-x-2\), (ii) \(x^{6}-3 x^{5}+6 x^{3}-3 x^{2}-3 x+2\), (iii) \(x^{5}-2 x^{4}-2 x^{3}+4 x^{2}+x-2\), (iv) \(x^{6}-2 x^{5}-4 x^{4}+6 x^{3}+7 x^{2}-4 x-4\), (v) \(x^{6}-6 x^{5}+12 x^{4}-6 x^{3}-9 x^{2}+12 x-4\).
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