Exercise 6, § 11

Let K be a field of characteristic p, n an integer not a multiple of p and N a splitting field of the polynomial $X^n - a$ of $K[X]$.

a) Show that the group Gal(N/K) is isomorphic to a subgroup of the group $\Gamma$ of matrices $\begin{pmatrix} x & y \\ 0 & 1 \end{pmatrix}$, where $x \in (\mathbf{Z}/n\mathbf{Z})^*$ and $y \in \mathbf{Z}/n\mathbf{Z}$. If $n$ is a prime number, the group Gal(N/K) is commutative only if $R_n(K) = K$ or if $a$ is the n-th power of an element of K. b) If $K = \mathbf{Q}$ and if $n$ is a prime number, show that if $a$ is not the n-th power of a rational number, then Gal(N/K) is isomorphic to the whole of $\Gamma$. c) If $K = \mathbf{Q}$, determine Gal(N/K) when $a$ is a square-free integer.