At a total loss with these Galois groups.

[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...?

 

[lippyka]

The Galois group of a field extension L:K is the set of isomorphisms from L to itself that fix K. The elements of the Galois group are functions from L to itself, and they form a group under composition.In this case, there are two intermediate fields, Q(\sqrt{3}) and Q(\sqrt{2}). The Galois group can be determined by finding all of the automorphisms of the extensions. An automorphism is an isomorphism that fixes the base field.One way to find the automorphisms is to look at the minimal polynomial. In this case, the minimal polynomial is x^4 - 10x^2 + 1. This polynomial has four roots, which correspond to the four elements of the Galois group. Specifically, these elements are the functions that send each of the four roots to any of the other three roots. These four functions form a group under composition, and this group is the Galois group of the extension.It is also possible to determine the Galois group by looking at the intermediate fields. Since there are two intermediate fields, there are two automorphisms of the extension that fix the base field. These two automorphisms are the functions that send each of the two intermediate fields to the other one. These two functions together with the identity form a group, which is the Galois group of the extension.