Different ways to prove $\sqrt p$ irrational for $p$ prime. I know this fact can be proved by contradiction(reductio ad absurdum) but please give proofs by different methods.

By Eisenstein's criterion, the polynomial $x^2-p$ is irreducible.

(To answer WimC's complaint below: Consider the splitting field of $x^2-p$ over $\mathbf Q_p$. It contains an element of valuation $1/2$, so it is a proper over-field of $\mathbf Q_p$, and therefore $x^2-p$ is irreducible over $\mathbf Q_p$ (and a fortiori over $\mathbf Q$)).