A cyclotomic field splits a finite group

Brauer induction expresses $[V]$ integrally as inductions of linear characters of elementary subgroups. Every such linear character $\lambda$ satisfies $\lambda (h{)}^m=1$, so it takes values in $F$ and its induced representation has an $F$-model. Thus $[V]$ is the scalar extension of a virtual $F$-representation.

By [F2], decompose that virtual $F$-representation as ${\sum }_i{c}_i[W_i]$ with the $W_i$ distinct irreducible $F$-representations. Base change preserves intertwiner spaces, so distinct $W_i{\otimes }_F\mathbb{C}$ have disjoint irreducible complex constituents. Each is semisimple, with positive constituent multiplicities. Since their signed sum is the single irreducible basis element $[V]$, exactly one summand occurs, its coefficient and the multiplicity of $V$ are both $1$, and it has no other constituent. Hence $V\cong W_i{\otimes }_F\mathbb{C}$ for that index. Every irreducible complex representation is therefore defined over $F$, so $F$ is a splitting field.

The exponent $m$ divides $\mathrm{\mid }G\mathrm{\mid }$, so $\mathbb{Q}({\zeta }_{\mathrm{\mid }G\mathrm{\mid }})$ contains every $m$-th root of unity. Applying the proved assertion to this field gives the final statement. ∎