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

• I
• kmitza

#### 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.

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.

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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

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.