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No, this is certainly not the case in general: In the paper "Sums of squares and the fields $\mathbb{Q}_{A_{n}}$" ( Journal of Algebra, 1994), John Thompson and I proved that for $n > 24$, the field generated by the complex character values of $A_{n}$ is $\mathbb{Q}[\{ \sqrt{\epsilon(p)p} \}]$, where $p$ runs through odd primes $p < n$ with $p \neq n-2$, and $\epsilon(p)$ is the sign with $p \equiv \epsilon(p)$ (mod $4$).