The constructible numbers form a field closed under square roots

Statement

The set of the constructible numbers is a subfield of $\mathbb{C}$. If $z$ is constructible, then the square roots of $z$ are constructible. Moreover, the field of constructible numbers is the smallest subfield of $\mathbb{C}$ that is closed under square roots.

Remarks

The proof of closure uses classical constructions: parallels and similar triangles give products and quotients, and a circle on a diameter of length $1 + x$ gives $\sqrt{x}$. Conversely, the intersection points of lines and circles have coordinates that come from the old ones by field operations and square roots. So a number is constructible if and only if it is obtained from $\mathbb{Q}$ by finitely many field operations and square roots. The constructible real numbers form an ordered field in which every positive element has a square root.

Explanations

L1 human

You can add, subtract, multiply and divide lengths with straightedge and compass, and you can take square roots. Nothing else is possible, because lines and circles only lead to equations of degree one or two.