I have two question regarding the Galois group of the splitting field of $x^4-3$.
Firstly I know the roots of this polynomial are $\sqrt[4]{3},w\sqrt[4]{3},w^2\sqrt[4]{3},w^3\sqrt[4]{3}$, where $w$ is the primitive 4th root of unity.
I also know the extension is Galois.
I happen to know as well that it is isomorphic to $D_8$.
Question 1) Finding the minimum polynomial of w
I had thought that the min. poly was found by noting its a root of $x^4-1=(x-1)(x^3+x^2+x+1)$, then as we know it's not 1 , it must be a root of $(x^3+x^2+x+1)$, but that gives the wrong degree of the extension.
I have an inkling that the minimum polynomial of a primitive nth root is the cyclotomic polynomial of the same n.
So here we'd have $\Psi_4=(x-w)(x-w^3)$, which has degree two , the correct degree.
Is that correct ?
Question 2) More of a general question on Galois automorphisms , but I thought of it when dealing with this example.
We must be able to send roots to roots of min. polynomials, so we can send $\sigma :\sqrt[4]{3}\rightarrow w\sqrt[4]{3} $, and $\tau: w \rightarrow w^2$.
But now, my question is, when we are constructing the group elements of of our Galois group , how do we use these mappings to construct all 8.
I have a few but am a little confused about what the rest should be :
$e: \sqrt[4]{3},w$
$\alpha: w\sqrt[4]{3},w$
$\gamma: \sqrt[4]{3},w^2$
$\beta: w\sqrt[4]{3},w^2$
How I found these was by applying tau and sigma, in different ways, but how can we find the rest ?
Best Answer
The more usual notation for a primitive fourth root of unity is $i$, where $i^2=-1$, rather than $w$, so I'll use that. So there is no automorphism mapping $w=i$ to $w^2=-i$. There is an automorphism $\tau$ mapping $\sqrt[4]3$ to itself and $i$ to $-i=i^3$, namely complex conjugation. There is also an automorphism $\sigma$ with $\sigma(i)=i$ and $\sigma(\sqrt[4]3)=i\sqrt[4]3$. These generate the (dihedral) Galois group $G$, with relations $\sigma^4=\tau^2=\text{id}$ and $\tau\sigma=\sigma^{-1}\tau$. The elements of $G$ are $\text{id}$, $\sigma$, $\sigma^2$, $\sigma^3$, $\tau$, $\tau\sigma$, $\tau\sigma^2$, $\tau\sigma^3$.