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 Theorem
The real number $x$ can be constructed using straightedge and compass if and only if there 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!