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Let $G$ be a finite group, and $g\in G$ an element of order $n$.

Let $\chi_1,\ldots,\chi_r$ be the irreducible characters of $G$.

Are there interesting conditions on $G$ and/or $g$ such that

$$\mathbb{Q}(\chi_1(g),\chi_2(g),\ldots,\chi_r(g)) = \mathbb{Q}(\zeta_n)$$

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    The characters of any symmetric group are integer valued, so it doesn't hold in general. – 2017-01-11
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    @Watson Good point! I've edited the question accordingly. – 2017-01-11
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    @user1952009 Note that $\chi_i(g)$ is in general not a root of unity. – 2017-01-11

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