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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.

Thanks in advance.

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    For a similar question where the product is replaced by the sum, see the MO question http://mathoverflow.net/questions/26832/de$g$ree-of-sum-of-algebraic-numbers. The article "Observations on primitive, normal, and subnormal elements of field extensions" by Steven Weintraub, appearing in Monatsh. Math. 162 (2011) pp. 239-244 might also be helpful.2011-06-20

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