Prove equality of number fields

[shabbado]

Hello everyone,

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

where Q = rationals

Any help would be appreciated.

Thanks

 

[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.

 

[Hurkyl]

Degree counting could work too.

 

[shabbado]

Degree counting could work too.

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.

 

[Hurkyl]

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

 

[shabbado]

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?

 

[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.