Norm principle for even K-groups of number fields

TL;DR

This paper studies the behavior of norm maps in algebraic even K-groups over number fields, establishing conditions for surjectivity and criteria for elements to be norms, primarily depending on real primes.

Contribution

It provides new results on the surjectivity of norm maps in even K-groups and introduces a criterion based on real primes for elements to be norms from extensions.

Findings

Norm maps are surjective in most cases.

A criterion based on real primes determines when elements are norms.

The results clarify the structure of even K-groups over number fields.

Abstract

We investigate the norm maps of algebraic even $K$-groups of finite extensions of number fields. Namely, we show that they are surjective in most situations. In the event that they are not surjective, we give a criterion in determining when an element in the even $K$-group of the base field comes from a norm of an element from the even $K$-groups of the extension field. This latter criterion is only reliant on the real primes of the base field.