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You are not logged in. #1 20090422 00:29:51
Finite fieldsAs 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. 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 (20090422 04:42:57) #2 20090423 17:54:24
Re: Finite fieldsAnother 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. 