# When the extensions Q(α,β) and Q(αβ) over Q are the same?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha\beta)$ over $\mathbb{Q}$ are the same.

I don't think it's productive to seek a complete characterization. In practice, you could do this by computing the characteristic polynomial of $\alpha \beta$ acting on a basis for $\mathbb{Q}(\alpha, \beta)$ and factor it. In special cases, if $\mathbb{Q}(\alpha, \beta)$ is Galois you might be able to explicitly write down the action of the Galois group on $\alpha \beta$. – Qiaochu Yuan Jun 20 '11 at 17:35
If $\gamma = \alpha \beta$, you're basically asking whether $\alpha \in {\mathbb Q(\gamma)}$ (if so, then $\beta = \gamma/\alpha$ is too, except in the case $\alpha = 0$ which I'll leave you to figure out). – Robert Israel Jun 20 '11 at 17:54