Example of extension $\mathbb{Q}(a,b) / \mathbb{Q}$ that is not simple? And more.

Question

I'm looking for a few examples of particular field extensions $\mathbb{Q}(\alpha,\beta) / \mathbb{Q}$.

(i) Is it possible to find a non-simple one? So I would like to show that there exists a field $\mathbb{Q}(\alpha,\beta)$ that cannot be written as $\mathbb{Q}(\alpha,\beta)=\mathbb{Q}(\gamma)$ for any $\gamma \in \mathbb{Q}(\alpha,\beta)$.
(ii) If (i) is not possible could I still find one such that $\mathbb{Q}(\alpha,\beta)\not = \mathbb{Q}(\alpha + \beta)$

I know about the Primitive element theorem , in particular that every finite separable extension is simple. The thing is that I don't know how to construct a non-finite or non-separable extension, if it even exists, so I guess (i) might not be possible.

For (ii), reading the Wikipedia page of the theorem, I saw that there exists only finitely many $\gamma := \alpha + c\beta$ with $c\in \mathbb{Q}$ that generate $\mathbb{Q}(\alpha , \beta)$, so there could actually be a case where $\alpha + \beta$ satisfies (ii).

Thank you for any answer you might have.

Answer 1

In general, non-finite extensions do exist. For example the extension $\mathbb{R}/\mathbb{Q}$ is going to be non-finite. But also the algebraic extension $\bar{\mathbb{Q}}/\mathbb{Q}$, where $\bar{\mathbb{Q}}$ is the algebraic closure, is non-finite. Both of these extensions are not primitive, but they are also not generated by adjoining two elements only.

To find such a field let's think about what you wrote here:

I know about the Primitive element theorem , in particular that every finite separable extension is simple. The thing is that I don't know how to construct a non-finite or non-separable extension, if it even exists, so I guess (i) might not be possible.

You already know that every finite, seperable extension is primitive. Now include your condition that we only want $2$ generators, i.e. $\mathbb{Q}(a,b)$. We know, that if $a,b$ are algebraic elements over $\mathbb{Q}$, then the extension is going to be finite and seperable (all algebraic extensions in characteristic $0$ are seperable). So, the only thing that we have left to do is adding non algebraic elements. For example, one could add variables, to obtain $\mathbb{Q}(x,y)$. Or, you could add $\pi$ and a variable: $\mathbb{Q}(x,\pi)$. Both of these extensions satisfy condition (i) and (ii) alike.

If you want to find fields where an algebraic extension satisfies (i), you need to stick to fields with characteristic $p>0$.

Edit: I just saw in the comments under your question that you are looking for number fields. The above discussion should make clear why this is not possible.