= Finite field cardinality is a prime power
{title2=$|F|=p^d$}
A <finite field> has positive <characteristic>, because its repeated sums of $1$ cannot all be distinct. By the <characteristic of an integral domain> theorem, this is a <prime number> $p$. Its multiples of $1$ form a copy of the <prime field> $\mathbb F_p$: a nonzero multiple $a1$ has inverse $b1$, where the <Bezout identity> $ab+cp=1$ supplies $b$.
View $F$ as a <vector space> over this <prime field>. Repeatedly extend a <linearly independent set> until it is maximal; the procedure terminates because $F$ is finite. Maximality makes it a <basis>. If the <dimension> is $d$, unique coordinates give exactly $p^d$ elements. Thus \b[the cardinality is a power of a single prime], rather than an arbitrary composite integer.
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