The Stacks project

Exercise 111.1.7. Suppose that $k$ is a field having a primitive $n$th root of unity $\zeta $. This means that $\zeta ^ n = 1$, but $\zeta ^ m\not= 1$ for $0 < m < n$.

1. Show that the characteristic of $k$ is prime to $n$.

2. Suppose that $a \in k$ is an element of $k$ which is not an $d$th power in $k$ for any divisor $d$ of $n$ for $n \geq d > 1$. Show that $k[x]/(x^ n-a)$ is a field. (Hint: Consider a splitting field for $x^ n-a$ and use Galois theory.)