Ok this might seem really trivial, but I'm not sure I'm doing it right.
If K,L are finite fields with K a subfield of L, then the extension L:K is finite.
Is it enough to know that since L is finite, as a K-vector space it must have a finite basis since at least L itself is a spanning set? Gosh I don't remember my linear algebra.
What if I want to find [L:K]? Suppose K has p^m elements and L has p^n elements. Then [L:K]= n/m? (the reason I figured was that if I do have the degree [L:K]=t and K has p^m elements, then L must have p^tm elements). How would I go about showing that? Is it just straightforward producing a basis?