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Give two numbers $a$ and $b$ which are algebraic over $\mathbb{Q}$ with $[\mathbb{Q}(a):\mathbb{Q}]=2$, $[\mathbb{Q}(b):\mathbb{Q}]=3$, but the degree of the minimal polynomial for $ab$ is smaller than $6$.

I have no idea how to approach. If you can't give me an answer, I'd appreciate a starting point suggestion. Thanks so much.

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    @Qiaochu: the OP was defaced for some reason. I've rolled back.2011-12-05

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Take $a=\zeta_3$, so $[\mathbb{Q}(a):\mathbb{Q}]=2$ and $b=\sqrt[3]{2}$. It follows that $ab=\zeta_3\sqrt[3]{2}$ which is a root of $X^3-2$.

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    @TomBrown: Yes.2011-12-05