Artin's induction theorem

Remarks

For the proof, let $\psi$ be a class function on $G$ orthogonal to all characters induced from cyclic subgroups. By Frobenius reciprocity, the restriction of $\psi$ to each cyclic subgroup is orthogonal to all its characters, so it is zero. Every element of $G$ lies in a cyclic subgroup, so $\psi = 0$. Hence the induced characters span the space of class functions over $\C$, and a rationality argument gives rational coefficients. Rational coefficients are needed in general. For example, let $V = \Z/2 \times \Z/2$ be the Klein four-group. Its cyclic subgroups are proper, so every character induced from one of them has even degree, and so has every integer combination of such characters. Hence the trivial character $1_V$, of degree $1$, is not an integer combination. But $1_V = \frac{1}{2}\sum_{C} \operatorname{Ind}_C^V 1_C - \frac{1}{2}\operatorname{Ind}_{1}^{V} 1$, with $C$ running over the three subgroups of order $2$. Brauer's theorem on induced characters reaches integer coefficients by using elementary subgroups instead of cyclic ones.

Going deeper

For a student

Artin used this to show that a power of every Artin L-function is a quotient of products of abelian L-functions, so that some power of it is meromorphic. Brauer's theorem later removed the power. The theorem is a statement about the rational character ring: tensored with $\Q$, the induction maps from cyclic subgroups are jointly surjective.