Characteristic of a field

Remarks

This is the characteristic of a ring applied to $K$. Since a field is an integral domain, the characteristic is zero or a prime. Example: in $\F_5$, $5 \cdot 1 = 0$ and $n \cdot 1 \ne 0$ for $1 \le n \le 4$, so $\operatorname{char} \F_5 = 5$. Non-example of positive characteristic: in $\Q$, $n \cdot 1 = n \ne 0$ for every $n \ge 1$, so no positive $n$ works and $\operatorname{char} \Q = 0$. The prime subfield of $K$ is isomorphic to $\Q$ if $\operatorname{char} K = 0$ and to $\F_p$ if $\operatorname{char} K = p$. If $\operatorname{char} K = p > 0$, then $(a + b)^p = a^p + b^p$ for all $a, b \in K$, which gives the Frobenius endomorphism.