As you know, a finite field always has order a power of a prime. I didnt really know how to prove this until I read Chapter 14 of John F. Humphreyss.
In fact, the result depends on just two results:
1 is proved by a combination of Lagrange and Sylow. By Lagrange, the order ofis a power of ; if there is also a prime dividing , then would have a Sylow -subgroup whose nonidentity elements would not have order a power of .
2 comes from the fact that the characteristic of a field is a prime; this means that every nonzero element of the field has order in the additive subgroup, and this implies that the additive group of the field is a -group.
Last edited by JaneFairfax (2009-04-21 06:42:57)
Another result about finite fields is that the multiplicative group of nonzero elements of a finite field is cyclic. This can be proved using a theorem about finite Abelian groups.