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I have two definitions of the said field.

And frankly I don’t see why one is equivalent to the other. It just doesn’t add up.

Let’s look at wikipedia’s definition.

In algebra, a Pythagorean field is a field in which every sum of two squares is a square.

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So, say there are $a,b \in K$ where $K $ is the Pythagorean closure (of $\mathbb{Q}$) These numbers must have the property $a^2+b^2=c^2$ for some $c \in \mathbb{Q}$. Yes? $K$ is a collection of all such numbers, so if I choose any two elements in $K$, square and sum them, I get a square number. Sure.

But how is this from Ian Stewart’s book the same?

The Pythagorean closure of $\mathbb{Q}$ is the smallest subfield $K \subseteq \mathbb{C}$ such that $z \in K \Rightarrow \pm \sqrt{z} \in K$

No, I don’t think they’re the same. How can the latter be deduced from the former? Or are Pythagorean FIELDS and CLOSURE different things? This is utterly confusing and I don’t understand it at all.

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