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Multiquadratic fields generated by characters of $A_n$

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 $π$-number is a positive integer whose prime factors belong to a set of odd primes $π:= \{p_1, p_2,\dots, p_t\}$. Let $K_π(A_n)$ be the field generated by the values of $A_n$-characters for even permutations whose orders are $π$-numbers. If $t\geq 2$, then we determine a constant $N_π$ with the property that for all $n> N_π$, we have $$K_π(A_n)=\mathbb{Q}\left(\sqrt{p_1^*}, \sqrt{p_2^*},\dots, \sqrt{p_t^*}\right).$$