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 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. ]]>