How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$?

Question

1) How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$ where $K=\mathbb C(X)$ ?

I would say that the minimal polynomial of $\sqrt[n]X$ on $K$ is $t^n-X\in K[t]$ which is separable, therefore $|\text{Gal}(K[\sqrt[n] X]/K)|=n$. Let define $X_1,...,X_n$ all the solutions. I want to define $\sigma_k: X_i\mapsto X_i^k$ and conclude that

$$\text{Gal}(K[\sqrt[n] X]/K)=\{\sigma _0,...,\sigma _{n-1}\}\cong \mathbb Z/n\mathbb Z$$

What do you think ?

2) How can I compute the galois group $\text{Gal}(\mathbb F_7[\alpha]/\mathbb F_7)$ where $\alpha$ is a root of $t^6-1$ ?

I don't know what to do here.

Answer 1

Hints:

1. This is a Kummer type root extension and thus Galois. You should observe that the (other) roots of $t^n-X$ are of the form $X_k=\zeta^k \root n\of X$, where $\zeta=e^{2\pi i/n}\in\Bbb{C}\subset K$ is a primitive root of unity of order $n$, and $k=0,1,2,\ldots,n-1$. Show that the Galois group is generated by the automorphism $\sigma$ that is fully determined by $\sigma(X_0)=X_1$.
2. What does Little Fermat tell you about the six non-zero elements of $\Bbb{F}_7$?