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I'm trying to prove that the characteristic of any field $F$ is either $0$ or a prime number, but I have no idea what to do. Help?

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    Hint: Example: Suppose char$(F)=12$. Let $0_F$ and $1_F$ be the additive and multiplicative identities of $F$. Let $a= 1_F+1_F+1_F$ and $b=1_F+1_F+1_F+1_F.$ Then $a\ne 0_F\ne b$ but by the Distributive Law $ab=\sum_{j=1}^{12}(1_F\cdot 1_F)=0_F$.2018-11-14

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