- #1
kmitza
- 17
- 4
- 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.
$$ \sigma^j = \tau^p = 1, \sigma^k\tau = \tau\sigma$$.
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.