Thread: Degree of an extension - easier way?

My book gives an example in which they calculate the degree of a field extension using . In particular it shows that . I feel like I might have an easier way to do it than the book does, and I was hoping someone could confirm that this works, or if it does not, why it does not.

Since , can we just say that since the minimal polynomial of is , , then similar to obtain , which gives us ?

The book first points out that we know that both 3 and 4 divide so it's at least 12, but then the minimum polynomial of is so (when is it less?) and so , yielding of course .

Thanks.

Originally Posted by AlexP

My book gives an example in which they calculate the degree of a field extension using . In particular it shows that . I feel like I might have an easier way to do it than the book does, and I was hoping someone could confirm that this works, or if it does not, why it does not.

Since , can we just say that since the minimal polynomial of is , , then similar to obtain , which gives us ?

The book first points out that we know that both 3 and 4 divide so it's at least 12, but then the minimum polynomial of is so (when is it less?) and so , yielding of course .

Thanks.

by Eisenstein's criterion, the minimal polynomial of over is but is the degree of the minimal polynomial of over not . so, the problem is that you didn't prove that is irreducible over
your textbook's approach is avoiding the above problem and i think it's a good approach.
the reason that is that is a zero of the polynomial and thus the degree of the minimal polynomial of over is at most

To clarify... Is it the fact that the minimal polynomial of has degree 4 in that gives us an upper bound of degree 4 on the minimal polynomial of in ?