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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.

For a ring $R$, its ring characteristic is the least positive integer $n$ with $n{1}_R=0$, or zero if no positive such integer exists. Equivalently it is the least positive integer satisfying $nr=0$ for every $r\in R$, because $nr=(n{1}_R)r$. This additive formulation also defines the characteristic of a rng when an identity is not assumed.

Let $R$ be an integral domain. Since $1_R\ne 0$, its characteristic is not one. If its positive characteristic were a composite integer $n=ab$ with $1<a,b<n$, minimality would give $a{1}_R\ne 0$ and $b{1}_R\ne 0$, butThis contradicts the absence of zero divisors. ThereforeThis is the characteristic of an integral domain theorem. If an integral domain is defined without requiring an identity, the same conclusion follows from the additive definition: for a hypothetical composite characteristic $n=ab$, choose $r$ with $ar\ne 0$, which exists by minimality of $n$. Then $(ar)(bs)=n(rs)=0$ for every $s$, forcing $bs=0$ for every $s$ and contradicting minimality again.