(Joint work with Kaique M.A. Roberto (IME-USP))

We will expand a fundamental tool in algebraic theory of quadratic forms to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory and Special Groups K-theory, developed by Dickmann-Miraglia. We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in order to provide a solution to Marshall's Signature Conjecture by Dickmann-Miraglia. Moreover, we will show how the extended version of Arason-Pfister Hauptsatz - presented in the previous talk "Multirings and applications to algebraic theory of quadratic forms, IV" - entails some interesting properties concerning K-theory and Marshall's conjecture.