# Galois Group

• April 5th 2010, 11:53 AM
mathman88
Galois Group
Find the Galois group of $f(x)=(x^3-2)(x^3-3)$.
• April 5th 2010, 02:30 PM
chiph588@
Quote:

Originally Posted by mathman88
Find the Galois group of $f(x)=(x^3-2)(x^3-3)$.

I could have this wrong, but here goes:

The splitting field for $f(x)$ is $K=\mathbb{Q}(\sqrt[3]2,\sqrt[3]3,\rho)$, where $\rho$ is a third root of unity.

We then have $[K:\mathbb{Q}]=18$

Since we can determine all automorphisms based on the roots of $f(x)$ and $\alpha\mapsto\alpha'$, where $\alpha'$ is another root of $m_{\alpha,\mathbb{Q}}(x)$, let's see where that takes us.
Define the automorphisms as follows:

$\sigma:\begin{cases} \sqrt[3]2\mapsto\rho\sqrt[3]2\\ \rho\mapsto\rho\\ \sqrt[3]3\mapsto\sqrt[3]3 \end{cases} \quad\quad \tau:\begin{cases} \sqrt[3]2\mapsto\sqrt[3]2\\ \rho\mapsto\rho^2\\ \sqrt[3]3\mapsto\sqrt[3]3 \end{cases} \quad\quad \gamma:\begin{cases} \sqrt[3]2\mapsto\sqrt[3]2\\ \rho\mapsto\rho\\ \sqrt[3]3\mapsto\rho\sqrt[3]3 \end{cases}$

Now note $|\sigma|=3,\; |\tau|=2,\; |\gamma|=3$, so $|<\sigma,\tau,\gamma>|=18= [K:\mathbb{Q}] = |\text{Gal}(K/\mathbb{Q})|$.

I'll leave it to you to see why $<\sigma,\tau> \cong S_3$.

So I'm thinking $\text{Gal}(K/\mathbb{Q}) = $, where $\gamma$ is a three-cycle.