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.