Sanity check finding a number $a$ such that $p(x)$ is the minimum polynomial of $a$ over $\Bbb{Q}$

Question

This question originates from Pinter's Abstract Algebra, Chapter 27, Exercise B4.

For each of the following polynomials $p(x)$, find a number $a$ such that $p(x)$ is the minimum polynomial of $a$ over $\Bbb{Q}$:

1. $x^2+2x-1$
2. $x^4+2x^2-1$
3. $x^4-10x^2+1$

Answer:

1. $-1+\sqrt{2}$
2. $\sqrt{-1+\sqrt{2}}$
3. $\sqrt{5+ 2\sqrt{6}}$

Correct?

Explanation:

1. $x^2+2x-1=0\implies x=-1\pm\sqrt{2}$
2. $x^4+2x^2-1=0\implies x^2=-1\pm\sqrt{2}\implies x=\pm\sqrt{-1\pm\sqrt{2}}$
3. $x^4-10x^2+1\implies x^2=5\pm2\sqrt{6}\implies x=\pm\sqrt{5\pm 2\sqrt{6}}$