Some natural question on subfield of Galois extension

Question

Let $\alpha,\beta\in \mathbb{\overline{Q}}$ and assume $\deg(\text{Irr}(\alpha,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q(\alpha)})=2$.

Then I strongly guess that $\mathbb{Q}(\alpha\beta)\ne \mathbb{Q}(\alpha,\beta)$ but I can't show it.

Is it true or false? If it is false, would you suggest me a counterexample?

Thanks!

Answer 1

It is wrong.

Take $\alpha=1+\sqrt{2}, \beta=1+\sqrt{3}$

$\mathbb Q(\alpha\beta) = \mathbb Q(1+\sqrt{2}+\sqrt{3}+\sqrt{6})=\mathbb Q(\sqrt 2 , \sqrt 3) = \mathbb Q(\alpha,\beta)$

For the only non-trivial equality, note that $\sqrt{2}+\sqrt{3}+\sqrt{6}$ is a primitive element of $\mathbb Q(\sqrt 2 , \sqrt 3)$ since it is not fixed by any of the three non-identitiy homomorphisms $\sqrt{2} \mapsto \pm \sqrt{2},\sqrt{3} \mapsto \pm \sqrt{3}$