Hermitian K-theory, Dedekind zeta-functions, and quadratic forms over rings of integers in number fields [Hermitian $K$-theory, Dedekind $zeta$-functions, and quadratic forms over rings of integers in number fields]

We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian K-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind zeta-functions for totally real abelian number fields. Our methods apply more readily to the examples of algebraic K-theory and higher Witt-theory, and give a complete set of invariants for quadratic forms over rings of integers in number fields.