Finite field cardinality is a prime power ($|F|=p^d$)

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 the cardinality is a power of a single prime, rather than an arbitrary composite integer.