Order of a finite division ring

Groups, Rings, Domains, Modules, etc, Galois theory
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tziaxri
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Joined: Mon Nov 09, 2015 4:32 pm

Order of a finite division ring

#1

Post by tziaxri »

Prove that the order of a finite division ring is power of a prime.
Tsakanikas Nickos
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Joined: Tue Nov 10, 2015 8:25 pm

Re: Order of a finite division ring

#2

Post by Tsakanikas Nickos »

Let \( \displaystyle D \) be a finite division ring. Then, by Wedderburn's Little Theorem, \( \displaystyle D \) is a finite field. It is a well-known result (from Galois Theory - we regard \( \displaystyle D \) as a vector space over its prime field, which has to be (isomorphic to) \( \displaystyle \mathbb{Z}_{p} \) ) that every such field contains \( \displaystyle p^{n} \) elements for some prime number \( \displaystyle p \) and some natural number \( \displaystyle n \).
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