I Generators of Galois group of ## X^n - \theta ##

[kmitza]

TL;DR Summary

If we have a polynomial $x^p - \theta$ for some prime p. Then we can show that it's Galois group has order p(p-1) then I want to prove what the group looks like described by generators and relations between them

As the summary says we have $f(x) = x^n - \theta \in \mathbb{Q}[x]$. We will call the pth primitive root $\omega$ and we denote $[\mathbb{Q}(\omega) : \mathbb{Q}] = j$. We want to show that the Galois group is generated by $\sigma, \tau$ such that
$$ \sigma^j = \tau^p = 1, \sigma^k\tau = \tau\sigma$$.

I know that the splitting field of $f$ is going to be $Q(t,\omega)$ and that the degree of this extension is going to be $[Q(t,\omega): :Q(\omega)][Q(\omega : Q)]$ where $t^p = \theta$, further as minimal polynomial of $\omega$ is going to be $p^{th}$ cyclotomic I have the second multiple being (p-1) and I can prove that the whole extension will have degree p(p-1). Now my idea is to define the morphisms as:
$$\sigma(t) = t, \sigma(\omega) = \omega^2$$ and $$\tau(t) = t\omega, \tau(\omega) = \omega$$
I can show that order of these two groups are p-1 and p but I don't know how to show that they generate my group.
I suspect that I am meant to construct the group as a semidirect product of $<\tau>$ and $<\omega>$ but I can't figure it out completely.

 

[Office_Shredder]

The order of the Galois group divides the degree of the extension, which you say you know is p(p-1). Do you know any numbers that must divide the order of the galois group?

Knowing the order of the Galois group helps a lot in figuring out what it is, because you can often rule out relationships existing between elements that you wouldn't have immediately guessed.

 

[kmitza]

The order of the Galois group divides the degree of the extension, which you say you know is p(p-1). Do you know any numbers that must divide the order of the galois group?

Knowing the order of the Galois group helps a lot in figuring out what it is, because you can often rule out relationships existing between elements that you wouldn't have immediately guessed.

Maybe I am mistaken but if I know the degree of the extension and I know it is Galois, don't I know that group is going to be of the order exactly the same as the degree? So I know that the order of the whole Galois group is p(p-1)? Now I know that I have a subgroup of order p which is immediately cyclic and I have a subgroup of order p-1 generated by $\sigma$ and as the two groups are of coprime order their intersection is the identity? Now from here I don't know what to do

 

[WWGD]

You say the subgroup of order p-1 is generated by a single element $\sigma$, so that it's also cyclic?
Edit: Because if you're correct, and both are cyclic, so is their direct product, and then you won't have the relations you described.