Show that ${a + b\pi : a, b \in \mathbb{Q}}$ does not qualify as a field extension

Question

How do I show that ${a + b\pi : a, b \in \mathbb{Q}}$ does not qualify as a field extension using $\sqrt{r}$?

Answer 1

I don't know what $\sqrt{r}$ is. But suppose that the above is a field. Then $\frac{1}{\pi}$ is in there. So in particular $\frac{1}{\pi}=a+b \pi$ for some $a,b \in \mathbb{Q},b \ne0$. This implies that $b\pi^2+a\pi-1=0$ which implies that $\pi$ is algebraic . This is a contradiction.