Exercise 5, § 17

Let $K$ be a field and $p$ its characteristic exponent. For every integer $n > 0$ and prime to $p$, denote by $\varphi_K(n)$ the degree over $K$ of $R_n(K) = K(\mu_n)$, where $\mu_n$ denotes the group of n-th roots of unity in an algebraic closure of $K$. For example $\varphi_Q(n) = \varphi(n)$ (V, p. 84, Th. 2); if $K$ is finite with $q$ elements, $\varphi_K(n)$ is the order of the residue class of $q$ in $(\mathbf{Z}/n\mathbf{Z})^*$ (V, p. 97, Cor.) ; if $K$ is algebraically closed, $\varphi_K(n) = 1$ for all $n$. a) Let $E$ be the relative algebraic closure of the prime field $K_p$ in $K$. Show that $\varphi_K(n) = \varphi_{F_p}(n)$. b) Suppose that $K$ is a finitely generated extension of $K_p$. Let $m$ be an integer $> 0$; then there exists an integer $N > 0$ divisible by every $n$ such that $\varphi_K(n) \leq m$. c) Let $s$ be an element of $GL_m(K)$ of finite order. Show that $s^N$ has order a power of $p$. (We have $s = tu = ut$, where $u$ has order a power of $p$ and $t$ has order $n$ prime to $p$. The element $t$ is root of a separable polynomial $X^n - 1$ as well as of its characteristic polynomial; deduce that the subalgebra $A$ of $M_m(K)$ generated by $t$ is etale, of degree $\leq m$, isomorphic to a product of extensions $R_d(K)$, where $d$ runs over certain divisors of $n$ including $n$ itself. Therefore $\varphi_K(n) \leq m$; use b.).