# Tower of fields and quadratic extensions

Suppose there is given a tower of fields in $$\mathbb{C}$$:

$$\mathbb{Q} =L_0 \subseteq L_1 \subseteq ... \subseteq L_n \supseteq \mathbb{Q}(\alpha)$$ with an $$\alpha \in \mathbb{C}$$

such that all extensions are quadratic:

$$[L_{j+1} : L_j] = 2$$.

Then I define the intersections with $$\mathbb{Q}(\alpha)$$ as follows:

$$M_j := L_j \cap \mathbb{Q}(\alpha)$$

and get a new tower of fields:

$$\mathbb{Q} =M_0 \subseteq M_1 \subseteq ... \subseteq M_n = \mathbb{Q}(\alpha)$$.

My question: Is the following claim correct:

It is either $$M_{j+1} = M_j$$ or $$[M_{j+1} : M_j] = 2$$.

Ian Stewart claims this in his book "Galois Theory" (4th edition, page 97) in a proof of one theorem. I don't know why this claim should be obvious. Probably this has something to do with the "Tower law": $$[M:K] = [M:L][L:K]$$ for $$K \subseteq L \subseteq M$$.

• Is $\mathbb{Q}(\alpha)$ assumed to be Galois? – Dean Young Apr 20 at 2:45