1
$\begingroup$

Let $K$ be the splitting field of $(x^2-2x-1)(x^2-2x-7)$. Find the Galois group $Gal(K/\mathbb{Q})$ and determine all intermediate subfields explicitly.

I have that $(x^2-2x-1)(x^2-2x-7) = (x-1-\sqrt{2})(x-1+\sqrt{2})(x-1-2\sqrt{2})(x-1+2\sqrt{2})$

Thus $K = \mathbb{Q}[\sqrt{2}]$.

I do not have much practice finding Galois groups, so could somebody outline explicitly how to find the Galois group and the intermediate subgroups?

$\endgroup$
0
2
$\begingroup$

You did a very good job determining the splitting field: $K=\mathbf{Q}(\sqrt{2})$. This is just a very simple example as you will see.

The extension $K/\mathbf{Q}$ is Galois and has degree two, so the Galois group is $\mathbf{Z}/2\mathbf{Z}$ (the only group, up to isomorphism, of order 2), consisting of the identity automorphism and the one interchanging $\sqrt{2}$ and $-\sqrt{2}$.

There are no non-trivial intermediate fields.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.