Prove that $\sqrt{2+\sqrt 3}$ is irrational

Question

Prove that $\sqrt{2+\sqrt3}$ is irrational. I can't seem to figure this one out.

Answer 1

Hint: $x\in \mathbb{Q} \implies x^2 \in \mathbb{Q}$

Answer 2

$\sqrt{2+\sqrt3}$ satisfies $x^4-4x^2+1=0$, which means that $\sqrt{2+\sqrt3}$ is an algebraic integer.

As shown in this answer, if $x$ is an algebraic integer, then $x\in\mathbb{Q}\implies x\in\mathbb{Z}$.

Since $1\lt\sqrt{2+\sqrt3}\lt2$, $\sqrt{2+\sqrt3}\not\in\mathbb{Z}$; therefore, $\sqrt{2+\sqrt3}\not\in\mathbb{Q}$.