# Show that the following extensions don’t qualify as a field extension

I'm assuming everyone's familiar with the concept of extension of $\mathcal{F}$ by $\sqrt{r}$.

How do I show that the following extensions don’t qualify as a field extension?

1. ${a + b\sqrt[3]{2} : a, b \in \mathbb{Q}}$
2. ${a + b\sqrt[4]{2} : a, b \in \mathbb{Q}}$
3. ${a + b\pi : a, b \in \mathbb{Q}}$

Any help would be much appreciated.

If $F$ is a field and $\alpha,\beta \in F$, then $\alpha\beta \in F$. Is this true for any of these sets?