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Let $(F,+,\cdot)$ be a field. The characteristic $\cha(F)$ of $F$ is commonly given by one of three equivalent definitions:

if $K$ is the prime subfield of $F$, then $\cha(F)$ is the size of $K$ if this is finite, and $0$ otherwise.
Note that the first definition also applies to arbitrary rings, and not just to fields.
The characteristic of a field (or more generally an integral domain) is always prime. For if the characteristic of $F$ were composite, say $mn$ for $m,n>1$, then in particular $mn$ would equal zero. Then either $m$ would be zero or $n$ would be zero, so the characteristic of $F$ would actually be smaller than $mn$, contradicting the minimality condition.
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