Hilbert's 11th problem

Let $K$ be an algebraic number field, i.e. a finite extension of $\mathbb{Q}$, and let $\mathcal{O}_K$ denote its ring of integers.

For $n \ge 1$, consider the quadratic forms

$$Q(x_1,\dots,x_n) = \sum_{i,j=1}^{n} a_{ij} x_i x_j, \qquad a_{ij} = a_{ji},$$

with coefficients $a_{ij} \in K$ (respectively $a_{ij} \in \mathcal{O}_K$), and assume $\det(a_{ij}) \neq 0$.

Call two such forms $Q, Q'$ in $n$ variables

• $K$-equivalent if there is $T \in \mathrm{GL}_n(K)$ with $Q'(x) = Q(Tx)$ for all $x$; - $\mathcal{O}_K$-equivalent if both have coefficients in $\mathcal{O}_K$ and there is $T \in \mathrm{GL}_n(\mathcal{O}_K)$ (that is, $T$ with entries in $\mathcal{O}_K$ and $\det T \in \mathcal{O}_K^{\times}$) with $Q'(x) = Q(Tx)$ for all $x$.

Give a complete classification of these forms up to $K$-equivalence and up to $\mathcal{O}_K$-equivalence: exhibit, for each $n$, a system of invariants of $Q$ attached to $K$ (and to $\mathcal{O}_K$) that can be computed from the coefficients $a_{ij}$ and that determines the equivalence class of $Q$, so that for any two given forms it can be decided whether they are equivalent, and describe which systems of invariant values actually occur.

Solve, in the same terms, the associated representation problem: for a given form $Q$ as above and a given element $c$ of $K$ (respectively of $\mathcal{O}_K$), decide from such invariants whether the equation

$$Q(x_1,\dots,x_n) = c$$

has a solution with $x_1,\dots,x_n \in K$ (respectively with $x_1,\dots,x_n \in \mathcal{O}_K$), and more generally determine the set of $c$ represented by $Q$ over $K$ and over $\mathcal{O}_K$; the case $c = 0$ with $(x_1,\dots,x_n) \neq (0,\dots,0)$ is included.