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## #1 2009-04-21 02:29:51

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)

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## #2 2009-04-22 19:54:24

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

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