How to prove an element doesn't belongs to a splitting field?

Question

Let $\alpha , \beta$ be the roots of $p(x)=x^4+3x^2+3$. I know that $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ have degree 4 over $\mathbb{Q}$

How can I prove that $\beta\notin\mathbb{Q}(\alpha)$?

Answer 1

We make use of the following lemma in Galois theory.

Lemma. Let $K$ be a field of characteristic $\neq 2$, $L\supset K$ a separable field extension and $\alpha,\beta\in L$ with $\alpha^2,\beta^2\in K$ but $\alpha,\beta\not\in K$. Then $K(\alpha)=K(\beta) \iff \alpha\beta\in K$.

In your case, we can calculate that the roots are $\pm\alpha=\pm\sqrt{\frac{-3+\sqrt{-3}}{2}}$ and $\pm\beta=\pm\sqrt{\frac{-3-\sqrt{-3}}{2}}$. We see that $\alpha$ and $\beta$ are quadratic over $\mathbf{Q}(\sqrt{-3})$. Now, $\alpha\beta=\sqrt{3}\not\in \mathbf{Q}(\sqrt{-3})$ (product of the roots is equal to the constant coefficient of $f$). We conclude by the preceding lemma that $\beta\not\in \mathbf{Q}(\alpha)$.