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I am trying to get my head around the proof of the following:

Suppose K is a finite field. With $p=charK, |K|=p^r$ where r is a positive integer.

I am supplied with the following proof:

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I do not understand how this proves what we are required to prove and I;m not satisfied with the last paragraph, can anyone offer some explanation?

Mark
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1 Answers1

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If $F$ is a subfield of $K$ then $K$ is an $F$-vector space. In the finite case, $r:=\dim_F K$ is certainly finite, which makes $K\cong F^r$. Now $\operatorname{char}K=p$ means precisely that $K$ contains a subfield $F$ of cardinality $p$. Hence $|K|=|F|^r=p^r$.

Hagen von Eitzen
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