Irrationality of √2 + √3

Suppose $r=\sqrt2+\sqrt3$ is rational. Since $r>0$ and
$$(\sqrt3-\sqrt2)(\sqrt3+\sqrt2)=3-2=1,$$
we get $\sqrt3-\sqrt2=\dfrac1r$, which is rational.

Then $\sqrt3=\dfrac12\left(r+\dfrac1r\right)$ would be rational.

But $\sqrt3$ is irrational. Indeed, if $\sqrt3=\frac pq$ with $p,q$ coprime, then $p^2=3q^2$, so $3\mid p^2$ and, $3$ being prime, $3\mid p$. Writing $p=3k$ gives $q^2=3k^2$, so $3\mid q$, contradicting coprimality.

Hence $\sqrt2+\sqrt3$ is irrational.