# Multiquadratic fields generated by characters of $A_n$

## TL;DR

This paper extends previous results on fields generated by character values of alternating groups, showing that multiquadratic fields can be generated by characters evaluated on elements with orders divisible only by ramified primes, for sufficiently large n.

## Contribution

It generalizes known results by characterizing fields generated by specific character values of $A_n$ related to ramified primes, confirming a conjecture of Thompson.

## Key findings

- For large n, $K_{ ext{pi}}(A_n)$ equals a multiquadratic field generated by square roots of primes in pi.
- The paper determines a constant $N_{ ext{pi}}$ beyond which the field equality holds.
- It confirms that suitable multiquadratic fields are generated by $A_n$-character values restricted to elements with orders divisible by ramified primes.

## Abstract

For a finite group $G$, let $K(G)$ denote the field generated over $\mathbb{Q}$ by its character values. For $n>24$, G. R. Robinson and J. G. Thompson proved that $$K(A_n)=\mathbb{Q}\left (\{ \sqrt{p^*} \ : \ p\leq n \ {\text{ an odd prime with } p\neq n-2}\}\right),$$ where $p^*:=(-1)^{\frac{p-1}{2}}p$. Confirming a speculation of Thompson, we show that arbitrary suitable multiquadratic fields are similarly generated by the values of $A_n$-characters restricted to elements whose orders are only divisible by ramified primes. To be more precise, we say that a $\pi$-number is a positive integer whose prime factors belong to a set of odd primes $\pi:= \{p_1, p_2,\dots, p_t\}$. Let $K_{\pi}(A_n)$ be the field generated by the values of $A_n$-characters for even permutations whose orders are $\pi$-numbers. If $t\geq 2$, then we determine a constant $N_{\pi}$ with the property that for all $n> N_{\pi}$, we have $$K_{\pi}(A_n)=\mathbb{Q}\left(\sqrt{p_1^*}, \sqrt{p_2^*},\dots, \sqrt{p_t^*}\right).$$