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Sorry for my bad English.

There is a famous problem of elementally number theory as follow;

Is an integer a sum of two rational squares iff it is a sum of two integer squares?

Now I want know about this generalization for a number field.

i.e. is the next Prop. true?

Let $K$ be any algebraic number field ,and $\mathscr{O}_K$ be ring of integers of $K$. For any $a\in \mathscr{O}_K$, $a$ can be written as $r^2+s^2$ for some $r,s\in K$ iff can be written as $b^2+c^2$ for some $b,c\in \mathscr{O}_K$.

Yos
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1 Answers1

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This is not true. For example, let $K=\Bbb{Q}(i)$ with $\mathcal{O}_K=\Bbb Z[i]$. Then $$ i = \left(\frac{1+i}{2}\right)^2 + \left(\frac{1+i}{2}\right)^2. $$ This shows $i$ is a sum of two squares in $\Bbb {Q}(i)$.

However, it is impossible to write $i$ as a finite sum of squares in ${\Bbb Z}[i]$ since $$ (a+bi)^2 = a^2 - b^2 + 2abi $$ has even imaginary part when $a$ and $b$ are in $\Bbb {Z}$. Thus any finite sum of squares in $\Bbb {Z}[i]$ has even imaginary part, so such a sum can't be equal to $i$.

Dietrich Burde
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