Square Root of Prime is Irrational

Theorem
The square root of any prime number is irrational.

Proof by Contradiction
Let $p$ be prime.

Suppose that $\sqrt p$ is rational.

Then there exist coprime integers $m$ and $n$ such that:

Any prime in the prime factorization of $n^2$ or $m^2$ must occur an even number of times (because they are squares).

Thus, $p$ must occur in the prime factorization of $n^2 p$ either once or an odd number of times.

Therefore, $p$ occurs as a factor of $m^2$ either once or an odd number of times, a contradiction in either case.

So $\sqrt p$ must be irrational.

Also see
The special case of $p = 2$ is a well-known mathematical proof.