As you know, a finite field always has order a power of a prime. I didn’t really know how to prove this until I read Chapter 14 of John F. Humphreys’s.

In fact, the result depends on just two results:

1 is proved by a combination of Lagrange and Sylow. By Lagrange, the order of

is 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 (2009-04-22 04:42:57)