= Solution
{c}
The roots-of-unity criterion for a <splitting field for finite group representations> says that the assumed $m$th <roots of unity> make $k$ a splitting <field> for $G$. Fix a multiplicative identification of the group $\mu_m(k)$ with the complex $m$th <roots of unity>. For a <p-regular element> $g$, its order divides $m$; the operator $\rho(g)$ is diagonalizable because $X^{|g|}-1$ has distinct roots in $k$. If its <eigenvalues>, counted with multiplicity, are $\lambda_1,\ldots,\lambda_d$, define the <Brauer character> by
$$
\chi_V(g)=\sum_{i=1}^{d}\widehat\lambda_i,
$$
where hats denote the chosen complex lifts. This defines a <class function> on the <p-regular elements>, not on arbitrary elements of $G$.
A <short exact sequence> can be represented by block triangular matrices, so the eigenvalue multiset is the union of those on its <submodule> and quotient. Thus <Brauer characters> are additive on <short exact sequences>. If $S_1,\ldots,S_t$ are the simple $kG$-modules, the <Jordan–Hölder theorem> gives
$$
\chi_V=\sum_{j=1}^{t}[V:S_j]\chi_{S_j}.
$$
We use the standard <Brauer–Nesbitt theorem> in its character form: over a splitting <field> the <Brauer characters> of the nonisomorphic <simple modules> are linearly independent over $\mathbb C$. Therefore
$$
\boxed{\chi_V=\chi_{V'}\iff[V:S_j]=[V':S_j]\text{ for every }j}.
$$
Both the modular splitting-field criterion and this independence theorem are the representation-theoretic results used here.
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