Let $K$ be some number field, $\mathcal O\_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\alpha \in \mathcal O\_K$, and consider the quantity $r\_{n,K}(\alpha)$, which denotes the number of solutions to the equation
$$
\alpha = x\_1^2 + x\_2^2 + \ldots + x\_n^2
$$
with $x\_1, x\_2, \ldots, x\_n \in \mathcal O\_K$.


When $K=\mathbb Q$ and $\mathcal O\_K=\mathbb Z$, the precise formulas for the computation of $r\_{n,K}(\alpha)$ were developed for all $n \leq 4$ (perhaps for other $n$'s as well, but I am not familiar with them). Are there any formulas for $r\_{n,K}(\alpha)$ when $[K:\mathbb Q] > 1$? I am especially interested in the cases when $n = 2, 3, 4$ and $K$ is a real quadratic extension of $\mathbb Q$.



Number of representations as sums of squares in rings of integers of number fields
There are some results known in the $n = 4$ case. In particular, in 1928 Gotzky (see Mathematische Annalen volume 100 pages 411-437) proved a formula for the number of representations of a totally positive integer $\alpha \in \mathcal{O}\_{K}$ as a sum of four squares of elements in $\mathcal{O}\_{K}$ for $K = \mathbb{Q}(\sqrt{5})$. In 1960, Harvey Cohn wrote a paper (in the American Journal of Mathematics, pages 301-322) determining formulas when $K = \mathbb{Q}(\sqrt{2})$ and $K = \mathbb{Q}(\sqrt{3})$. A 1961 paper of Cohn addresses the question of sums of three squares over $\mathbb{Q}(\sqrt{2})$ (where there is a formula) and $\mathbb{Q}(\sqrt{3})$ (where there might not be as clean a formula).


Kate Thompson, a recent Ph.D. student of Jonathan Hanke and Daniel Krashen, has been working on extending results about sums of four squares to other quadratic number fields.


