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We are looking for all subrings $R\neq 0$ of the field $(\mathbb{C},+,.)$ with the property that for every $x,y\in R$ there exists $z\in R$ such that $x^2+y^2=z^2$, and if $z=0$, then $x=y=0$.

It is obvious that $i\notin R$, and $\mathbb{R}$ enjoys the property but not $\mathbb{Q}$ and $\mathbb{C}$.

Can one characterize all such subrings?

Thanks in advance

M.H.Hooshmand
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    Start here https://en.wikipedia.org/wiki/Pythagorean_field – lhf Dec 15 '20 at 11:20
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    See also https://math.stackexchange.com/questions/116716/is-the-pythagorean-closure-of-mathbb-q-equal-to-the-field-of-constructible-nu – lhf Dec 15 '20 at 11:22

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