Chapter 19

Starting from Theorem \(19.11,\) show that $$\Pi_{F}(\ell)=\ell^{-1} \sum_{k \mid \ell} \mu(k) q^{\ell / k}$$ where \(\mu\) is the Möbius function (see \(\S 2.9)\).

How many irreducible polynomials of degree 30 over \(\mathbb{Z}_{2}\) are there?

This exercise develops an alternative proof for the existence of finite fields - however, it does not yield a density result for irreducible polynomials. Let \(F\) be a finite field of cardinality \(q,\) and let \(\ell \geq 1\) be an integer. Let \(E\) be a splitting field for the polynomial \(X^{q^{\ell}}-X \in F[X]\) (see Theorem 16.25), and let \(\sigma\) be the Frobenius map on \(E\) over \(F\). Let \(K\) be the subalgebra of \(E\) fixed by \(\sigma^{\ell}\) Show that \(K\) is an extension of \(F\) of degree \(\ell\).

Let \(E\) be an extension of degree \(\ell\) over a finite field \(F\) of cardinality \(q .\) Show that at least half the elements of \(E\) have degree \(\ell\) over \(F,\) and that the total number of elements of degree \(\ell\) over \(F\) is \(q^{\ell}+O\left(q^{\ell / 2}\right)\)

Let \(E\) be a finite extension of a finite field \(F,\) and suppose \(\alpha, \beta \in E,\) where \(\alpha\) has degree \(a\) over \(F, \beta\) has degree \(b\) over \(F,\) and \(\operatorname{gcd}(a, b)=1\) Show that \(\beta\) has degree \(b\) over \(F[\alpha]\), that \(\alpha\) has degree \(a\) over \(F[\beta],\) and that \(\alpha+\beta\) has degree \(a b\) over \(F\). Hint: consider the subfields \(F[\alpha], F[\beta], F[\alpha][\beta]=F[\alpha, \beta]=\) \(F[\beta][\alpha],\) and \(F[\alpha+\beta],\) and their degrees over \(F\)

Let \(E\) be an extension of degree \(\ell\) over a finite field \(F\). Show that for \(a \in F,\) we have \(\mathbf{N}_{E / F}(a)=a^{\ell}\) and \(\mathbf{T r}_{E / F}(a)=\ell a\).

Let \(E\) be a finite extension of a finite field \(F .\) Let \(K\) be an intermediate field, \(F \subseteq K \subseteq E .\) Show that for all \(\alpha \in E\) (a) \(\mathbf{N}_{E / F}(\alpha)=\mathbf{N}_{K / F}\left(\mathbf{N}_{E / K}(\alpha)\right),\) and (b) \(\operatorname{Tr}_{E / F}(\alpha)=\operatorname{Tr}_{K / F}\left(\operatorname{Tr}_{E / K}(\alpha)\right)\)

Let \(F\) be a finite field, and let \(f \in F[X]\) be a monic irreducible polynomial of degree \(\ell\). Let \(E=F[X] /(f)=F[\xi],\) where \(\xi:=[X]_{f}\) (a) Show that $$\frac{\mathbf{D}(f)}{f}=\sum_{j=1}^{\infty} \operatorname{Tr}_{E / F}\left(\xi^{j-1}\right) X^{-j}$$ (b) From part (a), deduce that the sequence of elements $$\operatorname{Tr}_{E / F}\left(\xi^{j-1}\right) \quad(j=1,2, \ldots)$$ is linearly generated over \(F\) with minimal polynomial \(f\). (c) Show that one can always choose a polynomial \(f\) so that sequence in part (b) is purely periodic with period \(q^{\ell}-1\).

Let \(F\) be a finite field, and \(f \in F[X]\) a monic irreducible polynomial of degree \(k\) over \(F\). Let \(E\) be an extension of degree \(\ell\) over \(F\). Show that over \(E, f\) factors as the product of \(d\) distinct monic irreducible polynomials, each of degree \(k / d\), where \(d:=\operatorname{gcd}(k, \ell)\).

Let \(E\) be a finite extension of a finite field \(F\) of characteristic \(p\). Show that if \(\alpha \in E\) and \(0 \neq a \in F\), and if \(\alpha\) and \(\alpha+a\) are conjugate over \(F\), then \(p\) divides the degree of \(\alpha\) over \(F\).

Let \(F\) be a finite field of characteristic \(p .\) For \(a \in F,\) consider the polynomial \(f:=X^{p}-X-a \in F[X]\) (a) Show that if \(F=\mathbb{Z}_{p}\) and \(a \neq 0,\) then \(f\) is irreducible. (b) More generally, show that if \(\operatorname{Tr}_{F / \mathbb{Z}_{p}}(a) \neq 0,\) then \(f\) is irreducible, and otherwise, \(f\) splits into distinct monic linear factors over \(F\).

Let \(E\) be a finite extension of a finite field \(F\). Show that every \(F\) -algebra automorphism on \(E\) must be a power of the Frobenius map on \(E\) over \(F\).

Show that for all primes \(p,\) the polynomial \(X^{4}+1\) is reducible in \(\mathbb{Z}_{p}[X]\). (Contrast this to the fact that this polynomial is irreducible in \(\mathbb{Q}[X]\), as discussed in Exercise \(16.49 .)\)

This exercise depends on the concepts and results in \(\S 18.6 .\) Let \(E\) be an extension of degree \(\ell\) over a finite field \(F\). Let \(\sigma\) be the Frobenius map on \(E\) over \(F\) (a) Show that the minimal polynomial of \(\sigma\) over \(F\) is \(X^{\ell}-1\). (b) Show that there exists \(\beta \in E\) such that the minimal polynomial of \(\beta\) under \(\sigma\) is \(X^{\ell}-1\) (c) Conclude that \(\beta, \sigma(\beta), \ldots, \sigma^{\ell-1}(\beta)\) form a basis for \(E\) over \(F\). This type of basis is called a normal basis.