Csdrhrt

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?

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.

Suppose $char(F) = m neq 0$.

By definition, $m$ is the smallest integer so that $underbrace{1 + ... + 1}_{m-times} = 0_F$

Suppose m is not prime and say $m = alpha.beta$, where $0 < alpha,beta < m$.

Therefore we can write:
$$
eqalign{
0_F &= underbrace{1 + ... + 1}_{m-times}\
&= (underbrace{underbrace{1 + ... + 1}_{alpha-times}) + ... + (underbrace{1 + ... + 1}_{alpha-times})}_{beta-times} tag{1}label{1}
}
$$

By closure property, $underbrace{1 + ... + 1}_{alpha-times} in F$. Let $underbrace{1 + ... + 1}_{alpha-times} = zeta tag{2}label{2}$ [It's important to not call it $alpha$. See Note]

There are 2 cases:

Case $zeta = 0_F$:

Since $alpha < m$ and $underbrace{1 + ... + 1}_{alpha-times} = 0_F, therefore char(F) = alpha$.

We arrive at a contradiction.

Case $zeta neq 0_F$:
$exists zeta^{-1} in F$
$$
eqalign{
therefore 0_F &= 0_F.zeta^{-1}\
&= (underbrace{zeta + ... + zeta}_{beta-times}).zeta^{-1} [by (ref{1}) and (ref{2})]\
&= underbrace{1 + ... + 1}_{beta-times}
}
$$

Since $beta < m, char(F) = beta$ and we again arrive at a contradiction.

Hence $m$ is a prime. QED

Note: We should not write $underbrace{1 + ... + 1}_{gamma-times} = gamma$, as $gamma$ is just an integer and not necessarily a member of the field $F$. Also this may lead to an erroneous conclusion, e.g. say since $char(F) = m$, if we write $underbrace{1 + ... + 1}_{m-times} = m = 0$, we can say $m = m.1 = 0$ and since $1 neq 0, therefore m = 0$.