Check if polynomial is minimal over $\mathbb{Q}$

Question

I want to determine the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb{Q}$.

I calculated the polynomial $x^4-10x^2+1$.

Now I used Eisensteins theorem to test if it is irreducible. It is not applicable in this form so I substituted $x$ with $x+1$.

$(x+1)^4 -10(x+1)^2 +1$

$x^4+4x^3 -4x^2 - 16x -8$

Here the prime $2$ divides all coefficient but the one of $x^4$ and by Eisensteins Theorem the polynomial is therefore irreducible over $\mathbb{Q}$.

But is this sufficient, I guess that it might still be reducible having coefficients of $\sqrt{2}$ or $\sqrt{3}$?

How can I test it for this numbers or am I on the wrong path and need to test it in another way?

Answer 1

A standard approach consists in going through the following steps. Set $E = \mathbb{Q}(\sqrt{2}, \sqrt{3})$.

1. Prove that $\sqrt{2} \in E$, so $K = \mathbb{Q}(\sqrt{2}) \subseteq E$.
2. Prove that $\sqrt{3} \in E$.
3. Prove that $\lvert K : \mathbb{Q} \rvert = 2$.
4. Prove that $\lvert E : K \rvert = \lvert K(\sqrt{3}) : K \rvert = 2$.
5. Prove that $\lvert E : \mathbb{Q} \rvert = 4$.
6. Conclude that the minimal polynomial of $\sqrt{2} + \sqrt{3}$ over $\mathbb{Q}$ has degree $4$.