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 $$K(A_n)=mathbb{Q}left ({ sqrt{p^*} : pleq n {text{ an odd prime with } p eq n-2}}right),$$ 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 $tgeq 2$, then we determine a constant $N_{pi}$ with the property that for all $n> N_{pi}$, we have $$K_{pi}(A_n)=mathbb{Q}left(sqrt{p_1^*}, sqrt{p_2^*},dots, sqrt{p_t^*}right).$$