Exercise 14, § 12

Let $K$ be a finite field, commutative or not, let $Z$ be its centre, $q$ the number of elements of $Z$ and $n$ the rank $[K : Z]$. a) If $E$ is a subfield of $K$ containing $Z$, show that $|E : Z|$ divides $n$. b) Let $x \in K$ be an element not belonging to $Z$; show that the number of distinct conjugates $yxy^{-1}$ of $x$ in $K^*$ is of the form $(q^n-1)/(q^d-1)$, where $d$ divides $n$ and is distinct from $n$ (consider in $K$ the set of elements permutable with $x$, and use a)). c) Deduce from b) that $q-1$ is divisible by the integer $\Phi_n(q)$ (partition the group $K^*$ into conjugacy classes and use the relation (6) of V, p. 82). d) Show that if $n > 1$, then $\Phi_n(q) > (q-1)^{\varphi(n)}$ (decompose $\Phi_n(X)$ in the field $\mathbf{C}$ of complex numbers). Deduce that we must have $K = Z$, in other words, that $K$ is necessarily commutative (Wedderburn's theorem).