About finite field

Darkseer

Let F be a field which contains finite elements. It is not difficult to prove that $|F|=p^n$ . And it can be shown that for every prime p there are such fields which contains p and $p^2$ elements. Is there a field containing $p^n$ elements, where $n>=3$ ?

**Posted:** Sat Oct 23, 2004 5:15 am

grobber

This is far from an open question. For each prime $p$ and each natural $n$ there is exactly one field with $p^n$ elements, and it's the splitting field of the polynomial $x^{p^n}-x$ over the field with $p$ elements.

**Posted:** Sat Oct 23, 2004 5:34 am

Darkseer

**Posted:** Sun Oct 24, 2004 1:14 am

grobber

Try google-ing "Galois theory".

Let $F$ be the splitting field of $f(x)=x^{p^n}-x$ over $F_p$ . Try to show that the roots of this polynomial are distinct, and form a subfield of $F$ , meaning that $F$ is, in fact, formed by these $p^n$ roots.

**Posted:** Sun Oct 24, 2004 4:21 am

zscool · Riemann Hypothesis Offline Joined: 22 Jun 2003 Posts: 482

if you can get your hands on: "A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics)"

by Ireland and Rosen, they have a nice section on finite fields (although they do not prove that fields of the same size are unique up to isomorphism) which provides a nice proof of the fact you stated (which would be take too much time to post to this board)

**Posted:** Sun Oct 24, 2004 11:12 am

Darkseer

Thanks for your informations.
I thought it had something to do with theories...but I was not sure.

**Posted:** Mon Oct 25, 2004 3:53 am