On the number of certain Galois extensions of local fields

TL;DR

This paper calculates the number of Galois extensions of local fields with Galois groups A_n and S_n, providing insights into their distribution and frequency.

Contribution

It introduces explicit formulas for counting Galois extensions of local fields with specific Galois groups A_n and S_n, advancing understanding in local field extension enumeration.

Findings

Derived formulas for counting A_n Galois extensions

Derived formulas for counting S_n Galois extensions

Enhanced understanding of local field extension distributions

Abstract

In this paper, we will calculate the number of Galois extensions of local fields with Galois group A_n and S_n.