Principal ideals in mod-$\ell$ Milnor $K$-theory

TL;DR

This paper investigates the structure of principal ideals in mod-$ ext{l}$ Milnor $K$-theory of fields, identifying generators and kernels related to norm varieties, extending previous work for the case when $ ext{l}=2$.

Contribution

It generalizes the description of principal ideals in mod-$ ext{l}$ Milnor $K$-theory for arbitrary primes, providing explicit generators and kernel characterizations.

Findings

Principal ideal generated by a symbol is the kernel of a specific $K$-theory map.

Provides generators for the annihilator of the ideal.

Extends known results from the case $ ext{l}=2$ to arbitrary primes.

Abstract

Fix a symbol $\underline{a}$ in the mod-$\ell$ Milnor $K$-theory of a field $k$, and a norm variety $X$ for $\underline{a}$. We show that the ideal generated by $\underline{a}$ is the kernel of the $K$-theory map induced by $k\subset k(X)$ and give generators for the annihilator of the ideal. When $\ell=2$, this was done by Orlov, Vishik and Voevodsky.