if $L/F$ a field extension, if b is algebraic over $F(a)$ and $b$ not algebraic over $F$, then $a$ is algebraic over $F(b)$. Example?

Question

I learned today that if $L/F$ a field extension, if b is algebraic over $F(a)$ and $b$ not algebraic over $F$, then $a$ is algebraic over $F(b)$. I wanted to find a small example to illustrate this. So I thought of $\sqrt{2}+\sqrt{3}$ which is algebraic over $\mathbb{Q}(\sqrt{2})$ (I think it is the root of $x^2-2 \sqrt{2}x-1$. I think that $\sqrt{2}+\sqrt{3}$ is not algebraic over $\mathbb{Q}$ (how do I show this?). So $\sqrt{2}$ is algebraic over $\mathbb{Q}(\sqrt{2}+\sqrt{3})$? Are there more easy examples?