algebraic number theory - sum of squares in ring of integers

To address the particularities of this question for number fields, the basic theorem is attributed to Hilbert, Landau and Siegel. First of all, any nonzero sum of squares in a number field has to be totally positive (that is, it is positive in all real embeddings). Hilbert (1902) conjectured that in any number field, a totally positive element is a sum of 4 squares in the number field. This was proved by Landau (1919) for quadratic fields and by Siegel (1921) for all number fields.

This sounds superficially like a direct extension of Lagrange's theorem, but there is a catch: it is about field elements, not algebraic integers as sums of squares of algebraic integers. A totally positive algebraic integer in a number field $K$ need not be a sum of 4 squares of algebraic integers in $K$. The Hilbert-Landau-Siegel theorem only says it is a sum of 4 squares of algebraic numbers in $K$.

For instance, in $mathbf{Q}(i)$ all elements are totally positive in a vacuous sense (no real embeddings), so every element is a sum of four squares. As an example,
$$
i = left(frac{1+i}{2}right)^2 + left(frac{1+i}{2}right)^2.
$$
This shows $i$ is a sum of two squares in $mathbf{Q}(i)$.
It is impossible to write $i$ as a finite sum of squares in ${mathbf Z}[i]$ since
$$
(a+bi)^2 = a^2 - b^2 + 2abi
$$
has even imaginary part when $a$ and $b$ are in $mathbf{Z}$. Thus any finite sum of squares in $mathbf{Z}[i]$ has even imaginary part, so such a sum can't equal $i$.
Therefore it is false that every totally positive algebraic integer in a number field is a sum of 4 squares (or even any number of squares) of algebraic integers.

Here are some further examples:

1. In $mathbf{Q}(sqrt{2})$, $5 + 3sqrt{2}$ is totally positive since
   $5+3sqrt{2}$ and $5-3sqrt{2}$ are both positive. So it must be a sum of at most four squares in this field by Hilbert's theorem, and with a little fiddling around you find
   $$
   5 + 3sqrt{2} = (1+sqrt{2})^2 + left(1 + frac{1}{sqrt{2}}right)^2 + left(frac{1}{2}right)^2 + left(frac{1}{2}right)^2.
   $$
   It is impossible to write $5 + 3sqrt{2}$ as a sum of squares in the ring of integers $mathbf{Z}[sqrt{2}]$ because of the parity obstruction we saw for $i$ as a sum of squares in $mathbf{Z}[i]$: the coefficient of $sqrt{2}$ in $5 + 3sqrt{2}$ is odd.




2. In $mathbf{Q}(sqrt{2})$, $sqrt{2}$ is not totally positive (it becomes negative when we replace $sqrt{2}$ with $-sqrt{2}$), so it can't be a sum of squares in this field. But in the larger field $mathbf{Q}(sqrt{2},i)$, everything is totally positive in a vacuous sense so everything is a sum of at most four squares in this field by the Hilbert-Landau-Siegel theorem. And looking at $sqrt{2}$ in $mathbf{Q}(sqrt{2},i)$, we find
   $$
   sqrt{2} = left(1 + frac{1}{sqrt{2}}right)^2 + i^2 + left(frac{i}{sqrt{2}}right)^2.
   $$

Hilbert made his conjecture on totally positive numbers being sums of four squares as a theorem, in his Foundations of Geometry. It is Theorem 42. He says the proof is quite hard, and no proof is included. A copy of the book (in English) is available at the time I write this as http://math.berkeley.edu/~wodzicki/160/Hilbert.pdf. See page 83 of the file (= page 78 of the book).

Siegel's work on this theorem/conjecture was done just before the Hasse-Minkowski theorem was established in all number fields (by Hasse), and the former can be regarded as a special instance of the latter.

Indeed, for nonzero $alpha$ in a number field $K$, consider the quadratic form $$Q(x_1,x_2,x_3,x_4,x_5) = x_1^2+x_2^2+x_3^2+x_4^2-alpha{x}_5^2.$$ To say $alpha$ is a sum of four squares in $K$ is equivalent to saying $Q$ has a nontrivial zero over $K$. (In one direction, if $alpha$ is a sum of four squares over $K$ then $Q$ has a nontrivial zero over $K$ where $x_5 = 1$. In the other direction, if $Q$ has a nontrivial zero over $K$ where $x_5 not= 0$ then we can scale and make $x_5 = 1$, thus exhibiting $alpha$ as a sum of four squares in $K$. If $Q$ has a nontrivial zero over $K$ where $x_5 = 0$ then the sum of four squares quadratic form represents 0 nontrivially over $K$ and thus it is universal over $K$, so it represents $alpha$ over $K$.) By Hasse-Minkowski, $Q$ represents 0 nontrivially over $K$ if and only if it represents 0 nontrivially over every completion of $K$.

Since any nondegenerate quadratic form in five or more variables over a local field or the complex numbers represents 0 nontrivially, $Q$ represents 0 nontrivially over $K$ if and only it represents 0 nontrivially in every completion of $K$ that is isomorphic to ${mathbf R}$. The real completions of $K$ arise precisely from embeddings $K rightarrow {mathbf R}$. For $t in {mathbf R}^times$, the equation
$x_1^2+x_2^2+x_3^2+x_4^2-t{x}_5^2 =0$ has a nontrivial real solution if and only if $t > 0$, so $Q$ has a nontrivial representation of 0 in every real completion of $K$ if and only if $alpha$ is positive in every embedding of $K$ into ${mathbf R}$, which is what it means for $alpha$ to be totally positive. (Strictly speaking, to be totally positive in a field means being positive in every ordering on the field. The orderings on a number field all arise from embeddings of the number field into $mathbf R$, so being totally positive in a number field is the same as being positive in every real completion.)

Siegel's paper is "Darstellung total positiver Zahlen durch Quadrate, Math. Zeit. 11 (1921), 246--275, and can be found online at http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN266833020_0011&DMDID=DMDLOG_0022.