Exercise 4, § 18

Let K be a finite field and A an algebraic K-algebra (Exercise 3).

a) Suppose that A is a field. Prove that A is commutative. (Let $x$ be a noncentral element of A. Consider the field $Z(x)$, where Z is the center of A, and prove that there exists an element $y\in A$ such that $yxy^{-1}=x^r$, with $x^r\not=x$. Deduce a contradiction by considering the subfield $K(x, y)$ of A.) b) Suppose that A does not contain any nonzero nilpotent element. Prove that the algebra A is commutative (use a) and Exercise 3).