The Hasse norm principle for $A_n$-extensions

TL;DR

This paper proves the Hasse norm principle for degree n extensions of number fields with Galois group A_n and demonstrates weak approximation for related norm one tori, advancing understanding in algebraic number theory.

Contribution

It establishes the Hasse norm principle for A_n-extensions of number fields and confirms weak approximation for the associated norm one tori, extending previous results.

Findings

Hasse norm principle holds for all n ≥ 5 in A_n-extensions

Weak approximation is valid for the associated norm one tori

Results apply to normal closures with Galois group A_n

Abstract

We prove that, for every $n \geq 5$, the Hasse norm principle holds for a degree $n$ extension $K/k$ of number fields with normal closure $F$ such that $\operatorname{Gal}(F/k) \cong A_n$. We also show the validity of weak approximation for the associated norm one tori.