# A genus formula for the positive \'{e}tale wild kernel

## TL;DR

This paper establishes a genus formula relating the positive étale wild kernel groups of a number field and its Galois extension, extending the understanding of their structure in algebraic number theory.

## Contribution

It introduces a genus formula for the positive étale wild kernel, connecting the kernel groups of a number field and its Galois extensions, a novel result in the field.

## Key findings

- Derived a genus formula linking kernel groups of extensions and base fields.
- Established the relationship between Galois group actions and kernel group orders.
- Enhanced the theoretical framework for studying étale wild kernels in number fields.

## Abstract

Let $F$ be a number field and let $i\geq 2$ be an integer. In this paper, we study the positive \'{e}tale wild kernel $\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}F$, which is the twisted analogue of the $2$-primary part of the narrow class group. If $E/F$ is a Galois extension of number fields with Galois group $G$, we prove a genus formula relating the order of the groups $ (\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}E)_{G}$ and $\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}F$.