Characteristic of a field is $0$ or prime
Is there any field of characteristic 4? Or any other composite number?
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No, a field $F$ can only have prime characteristic:
Let $a \neq 0 \in F$. If $(mn)a = 0$ then also $m(na)=0$. If $(na) \neq 0$ then $F$ has characteristic $\leq m$. If not, then $F$ has characteristic $\leq n$. Either way, the characteristic is $< mn$. This implies that a field must have characteristic prime or zero.
EDIT: Changed to additive notation.
This is a small variation on Fredrik Meyer's answer.
$0$ and $1 \neq 0$ are the additive and multiplicative identities in a field $\mathbb F$.
Thus the characteristic of field is either $0$ or a prime number.