Is there any field of characteristic 4 or any other composite number? 
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Characteristic of a field is $0$ or prime 

Is there any field of characteristic 4? Or any other composite number? 
 A: 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. 
A: This is a small variation on Fredrik Meyer's answer.
$0$ and $1 \neq 0$ are the additive and multiplicative identities in a field
$\mathbb F$.  


*

*If $1$, $1+1$, $1+1+1, \cdots$ are all distinct elements of $\mathbb F$,
then the characteristic of $\mathbb F$ is said to be $0$. Note that
$\mathbb F$ contains (at least) a countably infinite number of elements.

*If $1$, $1+1$, $1+1+1, \cdots$ are not all distinct, then if we
have for some $i$ and $j$,  $i < j$, that
$$
\underbrace{1+ \cdots + 1}_{i~\text{ones}} 
= \underbrace{1+ \cdots + 1}_{j~\text{ones}}
= \underbrace{1+ \cdots + 1}_{i~\text{ones}} 
+ \underbrace{1+ \cdots + 1}_{j-i~\text{ones}},
$$
we can conclude that $\underbrace{1+ \cdots + 1}_{j-i~\text{ones}} = 0$.
The smallest integer $N > 1$ such that $\underbrace{1+ \cdots + 1}_{N~\text{ones}} = 0$ is called the characteristic of the field.
(The possibility that $N$ could be $1$ is ruled out by the fact $1 \neq 0$
in a field).  $N$ must be a prime number because if $N$ were composite,
say $N = mn$ with $m,n > 1$, then from
$$
0 = \underbrace{1+ \cdots + 1}_{mn~\text{ones}} 
= \left(\underbrace{1+ \cdots + 1}_{m~\text{ones}}\right)
\times \left(\underbrace{1+ \cdots + 1}_{n~\text{ones}}\right),
$$
we get that at least one of $\underbrace{1+ \cdots + 1}_{m~\text{ones}}$
and  $\underbrace{1+ \cdots + 1}_{n~\text{ones}}$ is $0$, that is,
fewer than $N$ $1$'s sum to $0$ in contradiction of the definition of $N$.
Thus the characteristic of field is either $0$ or a prime number.
