Field Extension/Vector Space

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?

Re: Field Extension/Vector Space

If K and L are finite fields then they have p^n, q^m elements where p and q are primes and m and n are positive integers respectively. If L:K is a field extension then p = q and since L contains K, we have m > n, and thus p^n divides p^m. I.E. p^m/p^n = p^(m-n) (Remember m>n). and p^(m - n) is the index L : K, obviously finite.

You don't need to go off into Linear Algebra to get this result. But since you brought it up, we do now know that L is a finite dimensional vector space over K.

If we hadn't already known that we were working with finite fields, the argument might have been more delicate.