Irrationality of the square root of a prime

Statement

Let $p$ be a prime number. Then $\sqrt{p}$ is an irrational number.

Remarks

This generalizes the irrationality of the square root of 2. More generally, the square root of a natural number $n$ is irrational unless $n$ is a perfect square. See the roots of integers.

Explanations

L1 human

The numbers $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{7}$ are all irrational. The proof for $\sqrt{2}$ works for each prime: if $a^2 = p b^2$, then $p$ divides $a$, and then $p$ also divides $b$.