Say:

And let:

How do I show that is an element of the field extension:

(the problem is to show that this field extension is normal, which I can do if I can show the above)

Any help with this would be appreciated, thanks.

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- Mar 16th 2006, 09:16 AMDMTField Extensions
Say:

And let:

How do I show that is an element of the field extension:

(the problem is to show that this field extension is normal, which I can do if I can show the above)

Any help with this would be appreciated, thanks. - Mar 16th 2006, 01:20 PMrgep
We have and .

- Mar 16th 2006, 01:49 PMDMT
Thanks! Right in front of my face the whole time.

- Mar 17th 2006, 01:46 AMDMT
Quick additional question about another method for doing the same thing.

If I compared the fields:

and

And then showed that and have the same minimum polynomial, that would be enough to show that they are the same field, right? As would showing that the order of both extensions is 4 (which amounts to the same thing, right?). - Mar 17th 2006, 09:57 AMThePerfectHackerQuote:

Originally Posted by**DMT**

http://www.mathhelpforum.com/math-he...ead.php?t=1898

In general the answer is NO.

But you can do something else.

-----------------------------------

If you can show that then by the theorem of finite extension fields we have,

But since,

Thus,

Thus,

Thus,

But the trick and problem here is to show that the subfield relation is true. - Mar 18th 2006, 09:19 AMDMT
Thanks, that's about what I had in mind.

Here's a final question though ... what would be a basis for this field extension?

I can't seem to come up with a basis with only 4 elements. My first instinct was something like:

But this doesn't pick up multiples of . The only combos I can find that seem to cover all the elements of the field are too long. - Mar 18th 2006, 10:27 AMDMT
Never mind on that one ... it just hit me that since

then it's not really a problem.

The other thing I'm getting stuck with is the galois group for the field. It's supposed to be cyclic, but I seem to get the Klein 4-group instead.

I'm assuming the galois group is generated by the following two maps:

Okay, I see now that itself is order 4 and not order 2, and can generate a cyclic group of order 4, presumably the galois group. But that doesn't seem exactly right either since I don't really get in there.

Or rather, it seems but and seem to do weird things to the coeficient assuming my above basis, and not just change the signs of the root two terms (because of how they also change into ).

I'm wondering if my basis is wrong. Since

then maybe I can use in my basis instead of the term with ... would that work?

This is pushing the limit of my understanding of Galois theory which is what I am trying to learn.

What are the correct 4 maps for the Galois group?

Thanks.