Dedekind Zeta motives for totally real fields

Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $\pi^{n[k:\Q]}\zeta^*_k(1-n)$, where $\zeta^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$.