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

In summary, the conversation discusses the Galois group generated by two elements, sigma and tau, with certain properties. The order of the Galois group is p(p-1) and is a semidirect product of two cyclic subgroups of order p and p-1. The speaker is trying to figure out how to show that sigma and tau generate the group. The other person suggests using the order of the Galois group to rule out certain relationships between elements.
  • #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.
 
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  • #2
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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  • #3
Office_Shredder said:
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
 
  • #4
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.
 

1. What is the Galois group of a polynomial?

The Galois group of a polynomial is the group of automorphisms of its splitting field that fix the base field. In other words, it is the group of symmetries of the roots of the polynomial.

2. How do you find the generators of the Galois group of a polynomial?

To find the generators of the Galois group of a polynomial, one can use the Galois correspondence theorem which states that there is a one-to-one correspondence between subgroups of the Galois group and intermediate fields between the base field and the splitting field. By finding intermediate fields and their corresponding subgroups, one can determine the generators of the Galois group.

3. Can a polynomial have multiple Galois groups?

No, a polynomial can only have one Galois group. However, different polynomials can have the same Galois group.

4. What is the significance of the Galois group of a polynomial?

The Galois group of a polynomial provides important information about the polynomial and its roots. It can determine whether a polynomial is solvable by radicals, and it can also be used to classify polynomials into different equivalence classes.

5. How does the degree of a polynomial relate to its Galois group?

The degree of a polynomial is related to the order of its Galois group. Specifically, the degree of a polynomial is equal to the product of the orders of the generators of its Galois group.

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