Examples where minimal polynomial is smaller over a simple extension.

Question

I was working on some intuition-building exercises for Galois theory and ran into a problem that was a bit annoying. Are there any easy-to-check examples of algebraic numbers $a,b$ such that $[\mathbb{Q}(a,b) : \mathbb{Q}(a)] < [\mathbb{Q}(b) : \mathbb{Q}]$ with $a \not\in \mathbb{Q}(b)$ and $b \not\in \mathbb{Q}(a)$? I think I've got a working example, but I went through several that didn't work first and I had to cheat using intuition to get any reasonable candidates. Is there some systematic approach I should have used for finding such $a,b$? Is there a classic example I should keep in mind?