Find the Galois group of $x^4-12x^2+8x$

Question

I want to find the Galois group of $x^4-12x^2+8x$. What is the standard method to do this?

Is it to find resolvant cubic and its discriminant? If it is a perfect -$A_3$. If not- $S_3$?

Many thanks

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Edit: Let $f(x)=x^4-12x^2+8x=xg(x)$ where $g(x)=x^3-12x+8$ is irreducible by Eisenstein using $p=2$

Answer 1

I suppose you want to find the Galois group of the splitting field of this polynomial over the rational numbers $\mathbb{Q}$. There is a general method to find the Galois group of the splitting field of a third degree polynomial.

Let $L$ be the splitting field of the irreducible polynomial of degree 3 over $\mathbb{Q}$. You can show that if $\sqrt{\Delta} \in \mathbb{Q}$ we have $[L:\mathbb{Q}]=3$, so the Galois group is the cyclic group of order 3. Otherwise, if $\sqrt{\Delta}\notin \mathbb{Q}$ we get $[L:\mathbb{Q}]=6$ and therefore Galois group $S_3$.