Thread: 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?

Originally Posted by bleys

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.


Yes, it is enough.


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?

If you don't remember, as you say, your linear algebra, and it seems you really don't, then it could

prove to be a little difficult to explain this.

You better grab some algebra book (Dummit&Foote, Hungerford, etc.) and readi it there.

Tonio

Ps. By the way, it is true that , and from this oine can "guess" that it must be that m divides n...

oh it 'seems I really don't'; how insightful of you to deduce that from a sentence that even happens to be correct? I don't need your attitude in every post, tonio. A simple 'it's enough, but brush up on some linear algebra' or actual explanation is enough.

Originally Posted by bleys

oh it 'seems I really don't'; how insightful of you to deduce that from a sentence that even happens to be correct? I don't need your attitude in every post, tonio. A simple 'it's enough, but brush up on some linear algebra' or actual explanation is enough.

Fair enough, Bleys: I was just pointing out what you wrote, without any sarcasm, scoffing or whatever, in

order to make it plain to you that it'll be too long (at least for me) to explain you in this forum what's going on.

Don't worry, I won't piss you any more in the future with my lousy intents to help you out.

Tonio

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.