Z/pZ is a field for p prime

Statement

Let $n \ge 2$ be an integer. The ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ is a field if and only if $n$ is a prime number.

Remarks

If $n = a b$ with $1 < a, b < n$, then the classes of $a$ and $b$ are nonzero and their product is $0$. So $\mathbb{Z}/n\mathbb{Z}$ is not an integral domain. If $n = p$ is prime and $p \nmid a$, then Bézout's identity gives $a x + p y = 1$ for some integers $x, y$. So $x$ is an inverse of $a$ modulo $p$. The field $\mathbb{Z}/p\mathbb{Z}$ is often written $\mathbb{F}_p$. It is the smallest finite field of characteristic $p$.

Explanations

L1 human

Modulo $5$, every nonzero class has an inverse: $1 \cdot 1 = 1$, $2 \cdot 3 = 6 \equiv 1$ and $4 \cdot 4 = 16 \equiv 1$. Modulo $6$, the class of $2$ has no inverse, because $2 x$ is always even. Division works modulo $n$ exactly when $n$ is prime.