Definition 9.5.1. The characteristic of a field $F$ is $0$ if $\mathbf{Z} \subset F$, or is a prime $p$ if $p = 0$ in $F$. The prime subfield of $F$ is the smallest subfield of $F$ which is either $\mathbf{Q} \subset F$ if the characteristic is zero, or $\mathbf{F}_ p \subset F$ if the characteristic is $p > 0$.
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