The well-known Gauß-Wantzel Theorem states that a real number $ x$ can be constructed using straightedge and compass *only if* the minimal polynomial of $ x$ (over the field $ \mathbf Q$ ) has degree of form $ 2^n$ , $ n \in \mathbf N$ .

Is is a corollary of a more general theorem, named “Wantzel Theorem” in French, which (under the form I know) states that :

Wantzel TheoremThe real number $ x$ can be constructed using straightedge and compass

if and only ifthere exists a sequence of commutative fields $ L_0 \subset L_1 \subset … \subset L_n$ such that:

- $ L_0 = \mathbf Q$
- $ x \in L_n$
- For all $ i = 1, …, n$ , $ [L_i : L_{i-1}] = 2$

I wonder whether in the latter theorem, condition 2 could be replaced by $ L_n = \mathbf Q[x]$ .

Of course, this constraint 2′ implies constraint 2, so we have one implication.

To get the other implication, I assume I have a sequence $ L_0 \subset L_1 \subset … \subset L_n$ matching conditions 1, 2 and 3, and I set for $ i = 0, …, n$ , $ L’_i = L_i \cap \mathbf Q[x]$ . Then I need to prove that for $ i=1, …,n$ , $ [L’_i : L’_{i-1}] \le 2$ .

My question is: is the latter statement true?

In a more general way, **if $ L$ is a finite extension of field $ K$ , and $ M$ is another field, what can we say about $ [L \cap M : K \cap M]$ **?

Thanks in advance!