# Characteristic of a field.
Let $K$ be a field. Then take the multiplicative identity $1\in K$. If we repeatedly add $1$ to itself, either there exists some positive integer $n$ where $n\cdot 1 = 1 + 1 + \cdots + 1 =0$, or $n\cdot 1$ is never zero. In the former case, we have further that the smallest such positive integer $n$ must be a prime number, and we say $K$ is of *positive characteristic*; and in the latter case, we say $K$ has *characteristic zero*.
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Another way to think about this, consider the ring homomorphism $$
\phi: \mathbf{Z} \to K, \phi(1) = 1.
$$and this map is either injective or not injective.
When it is injective, then $\ker\phi = (0)$, which is the situation when $K$ is of characteristic zero. Here, the field $K$ contains a copy of the integers $\mathbf{Z}$, and since $K$ is a field, it contains a copy of $\mathbf{Q}$.
When it is not injective, then $\ker\phi =n\mathbf{Z}$ for some $n$. We now claim $n$ is a prime. Suppose not, that $n=a b$ for some $a,b > 1$. Then observe that $$
a b =\underbrace{ (1+\cdots + 1)}_{a}\underbrace{(1+\cdots +1)}_{b}=0.
$$But since $K$ is a field, $K$ is an integral domain. So either $a=0$ or $b=0$. In either case $\ker\phi = a\mathbf{Z}$ or $b\mathbf{Z}$ where $a,b < n$, a contradiction.
So every field contains either a copy of $\mathbf{Z}$ or $\mathbf{Z}/p\mathbf{Z}$ for some prime $p$.
B / 7 2024