Milnor-Witt K-theory and Witt K-theory of a field

TL;DR

This paper explores the structure of Milnor-Witt and Witt K-theories of fields, providing explicit computations and an elementary proof of an isomorphism related to Witt rings in characteristic two.

Contribution

It offers new elementary proof techniques for the isomorphism between Witt K-theory and the Rees algebra of the fundamental ideal in characteristic two.

Findings

Milnor-Witt K-theory computations for fields

Elementary proof of Witt K-theory isomorphism in characteristic two

Connection established via Kato's solution to Milnor's conjecture

Abstract

We recall some basic computations in the Milnor-Witt K-theory of a field, following Morel. We then focus on the Witt K-theory of a field of characteristic two and give an elementary proof of the fact that it is isomorphic as a graded ring to the Rees algebra of the fundamental ideal of the Witt ring of symmetric bilinear forms using Kato's solution to Milnor's conjecture on quadratic form.