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 .