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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.

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Hint: Are any of these sets closed under multiplication?

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

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  • $\begingroup$ I'm sorry I'll need more help than that. $\endgroup$ – HKT Apr 7 '17 at 0:29

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