# Prove equality of number fields

1. Dec 1, 2008

Hello everyone,

I need to prove that Q[i + sqrt(2)] = Q[squrt(2)]

where Q = rationals

Any help would be appreciated.

Thanks

2. Dec 1, 2008

### Dick

Prove i+sqrt(2) is in Q[sqrt(2)]. That's easy. Then prove i and sqrt(2) are in Q[i+sqrt(2)]. That's a little harder, but not much.

3. Dec 1, 2008

### Hurkyl

Staff Emeritus
Degree counting could work too.

4. Dec 1, 2008

thank you for taking the time to reply.

With degree counting, would that be counting the bases? Unforunately my professor did not explain this well and I am having a difficult time finding information regarding this subject.

5. Dec 1, 2008

### Hurkyl

Staff Emeritus
I'm not sure what you mean by "counting the bases". I'm referring to looking at the degrees of various field extensions.

6. Dec 1, 2008

a couple of things...

How exactly do I go about find the various field extensions and once I do, how does this help me prove equality?

7. Dec 1, 2008

### Dick

I'm not really sure what Hurkyl is up to, but just try the direct approach. Show Q[i+sqrt(2)] is a subset of Q[sqrt(2)] and conversely.