Exercise 2, § 8

Let $K$ be a field and $A$ a $K$-algebra of finite degree $n$. For each integer $m$ relatively prime to $n$ we have $A^{*m} \cap K^* = K^{*m}$; in other words, if $x \in K^*$ and if there exists an element $y$ of $A$ such that $x = y^m$, then there exists an element $z$ in $K$ such that $x = z^m$.