91Ó°ÊÓ

Integral root theorem. Let \(f(t)=t^{n}+\cdots+a_{0}\) be a polynomial of degree \(n \geq 1\) with integer coefficients, leading coefficient 1 , and \(a_{0} \neq 0 .\) Show that if \(f\) has a root in the rational numbers, then this root is in fact an integer, and that this integer divides \(a_{\mathrm{e}}\).

Show that the discriminant of a polynomial is 0 if and only if the polynomial has a root of multiplicity \(>1\). (You may assume that the polynomial has cocflicicnts in an algebraically closed field.)

(a) Let \(K\) be a subfield of a field \(E\), and \(\alpha \in E\). Let \(J\) be the set of all polynomials \(f(t)\) in \(K[t]\) such that \(f(\alpha)=0\). Show that \(J\) is an ideal. If \(J\) is not the zero ideal, show that the monic generator of \(J\) is irreducible. (b) Conversely, let \(p(t)\) be irreducible in \(K[t]\) and let \(\alpha\) be a root. Show that the ideal of polynomials \(f(t)\) in \(K[t]\) such that \(f(x)=0\) is the ideal generated by \(p(t)\).

Let \(W_{n}\) be the set of primitive \(n\) -th roots of unity in \(\mathbf{C}^{*}\). Define the \(n\) -th cyclotomic polynomial to be $$ \Phi_{n}(t)=\prod_{\zeta \in w_{v}}(t-\zeta) . $$ (a) Prove that \(t^{n}-1=\prod_{d \mid n} \Phi_{d} d t\). (c) Let \(p\) be a prime number and let \(k\) be a positive integer. Prove $$ \Phi_{p}(t)=\Phi_{p}\left(t^{p^{k+2}}\right) \quad \text { and } \quad \Phi_{p}(t)=t^{p-1}+\cdots+1 . $$ (d) Compute explicitly \(\Phi_{n}(t)\) for \(n \leqq 10 .\)

Let \(K\) be a finite ficid with \(q\) elements. Let \(\varphi: K \rightarrow K\) be any function of \(K\) into itself, Show that there exists a polynomial \(f\) over \(K\) such that \(\varphi(x)=f(x)\) for all \(x \in K\)