# At a total loss with these Galois groups.

1. Oct 31, 2009

### futurebird

I've been asked to match some galois groups with structures like:

Z_2, Z_3, Z_2 X Z_2 ...etc.

And I'm very much lost. I know the galois group for a field extension L:K is the set of isomorphism that fix F. These form a group with function composition.

OK. But how do I find these isomorphism. The book has one example and they gloss over that step... my instructor only did it for a cases where it was really obvious that it was conjugation. So, I'm feeling lost.

One of the problems is a little like this:

$$Q(\sqrt{3}, \sqrt{2}): Q$$

$$Q(\sqrt{3}, \sqrt{2})= Q(\sqrt{3}+ \sqrt{2})$$

So the minimal polynomial is

$$x^2 = 5 + 2\sqrt{6}$$

$$x^4 -10x^2 + 1$$

The degree is 4.

Here the intermediate fields are $$Q(\sqrt{3}), Q (\sqrt{2})$$ ... but are there more?

in any case how do I find the elements in the Galois group ...

The identity... and...?

Last edited: Oct 31, 2009