Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Finite fields

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 (2009-04-21 06:42:57)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Finite fields

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.

http://z8.invisionfree.com/DYK/index.php?showtopic=835