The classical Gauss problem of determining all imaginary quadratic number fields of class number one has an analogue involving finite motivic cohomology groups attached to the ring of integers $$o_F$$ in a totally real number field $$F$$ . In the classical situation, the value of the zeta function of $$F$$ at $$0$$ can be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the class number. For a totally real field $$F$$ and an even integer $$n \ge 2$$ , the value of the zeta-function of $$F$$ at $$1-n$$ can also be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the order $$h_n(F)$$ of the motivic cohomology group $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ . This order is always divisible by $$2^d$$ , where $$d$$ is the degree of $$F$$ . We determine all totally real fields $$F (\ne \mathbb{Q })$$ and all even integers $$n \ge 2$$ , for which the quotient $$\frac{h_n(F)}{2^d}$$ is equal to 1. There are no fields for $$n \ge 6$$ , there is only the field $$\mathbb{Q }(\sqrt{5})$$ for $$n = 4$$ , and there are 11 fields for $$n=2$$ . The motivic cohomology groups $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ contain the canonical subgroups $$WK^\mathcal M _{2n-2}(F)$$ , called motivic wild kernels, which are analogous to Tate–Shafarevic groups. For even integers $$n \ge 4$$ , there is again only the case $$n = 4$$ and the field $$\mathbb{Q }(\sqrt{5})$$ , for which the motivic wild kernel $$WK^\mathcal M _{2n-2}(F)$$ vanishes. However, for $$n=2$$ , among the totally real fields $$F$$ of degrees between $$2$$ and $$9$$ , we found 21 of them, for which $$WK^\mathcal M _{2}(F)$$ vanishes, the largest degree being $$5$$ . A basic assumption in our approach is the validity of the $$2$$ -adic Main Conjecture in Iwasawa theory for the trivial character, which so far has only been proven for abelian number fields (by A. Wiles).