# Pythagorean closure

Im reading the book "Galois Theory" by Ian Stewart $$(4$$th Edition$$)$$. Here the author defines the Pythagorean closure as follows:

Definition:The Pythagorean closure $$\mathbb{Q}^{PY}$$ of $$\mathbb{Q}$$ is the smallest subfield $$K \subseteq \mathbb{C}$$ with the property $$z \in K \Rightarrow \pm\sqrt{z} \in K$$.

A few pages later he says without further explanation:

Suppose $$\alpha \in \mathbb{Q}^{PY}$$. Then by definition there is a tower: $$\mathbb{Q} =L_0 \subseteq L_1 \subseteq ... \subseteq L_n \supseteq \mathbb{Q}(\alpha)$$ such that $$[L_{j+1} : L_j] = 2$$ for all $$j$$.

I understand that each quadratic adjunction has degree $$2$$. But for me it is not obvious why the existence of such tower follows from the definition. How can it be constructed?

New contributor
Philipp is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• The field consisting of the union of all such towers is closed under taking square roots. – Lord Shark the Unknown Feb 10 at 20:43
• What means the union of towers? Why is this a field? – Philipp Feb 10 at 22:11