Multiquadratic fields generated by characters of $A_n$

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 $$K(A_n)=\mathbb{Q}\left(\{\sqrt{p^{\ast }}:p\le n\text{ an odd prime with }p\ne n-2\}\right),$$ where $p^{\ast }:=(-1{)}^{\frac{p-1}{2}}p$. 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 $t\ge 2$, then we determine a constant $N_{\pi }$ with the property that for all $n>N_{\pi }$, we have $$K_{\pi }(A_n)=\mathbb{Q}\left(\sqrt{p_1^{\ast }},\sqrt{p_2^{\ast }},\dots ,\sqrt{p_t^{\ast }}\right).$$