TITLE: An unusual degree of a minimal polynomial
QUESTION [5 upvotes]: 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.

REPLY [10 votes]: 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 $X^3-2$.