# Characteristic of a field is $0$ or prime

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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## 1 Answer

Hint: Show that if the characteristic of $F$ were a composite number, say $n=ab$, then $F$ would have zero-divisors (since $n=0$...). Then show that a zero-divisor cannot be a unit unless $0=1$, which is not the case for fields.

Alternate Hint: Look at the unique homomorphism $\phi:\mathbb{Z}\to F$, defined by $\phi(0)=0_F$, $\phi(1)=1_F$, $\phi(2)=1_F+1_F$, etc. and note that its image, being a subring of a field, must be an integral domain.

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OK...I'm not very skilled at proofs, so can you see if there are any holes with this? I'm using the 1st Hint... Proof: Consider, for contradiction, that the field is a composite number. (still working) –  pigishpig Nov 28 '11 at 5:33