We have two numbers $a$ and $b$ algebraic over a field $K$, such that $a^2,b^2\in K$ but $a,b,ab\not\in K$. I have to characterize the galois group of $K[a,b]/K$
I think that I would be done if I were able to show that $a\not\in K[b]$, because in that case the galois group would be $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$. Am I right? How to prove that $a\not\in K[b]$, providing that it is true?