We have and .
Quick additional question about another method for doing the same thing.
If I compared the fields:
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?).
Your question is a very interesting one. I once asked a more general question on the forumOriginally Posted by DMT
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 the trick and problem here is to show that the subfield relation is true.
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.
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?