Over $\mathbb{C}$, a cyclic group of order $n$ has exactly $n$ irreducible representations up to equivalence, represented by the characters $g\mapsto \lambda$ with ${\lambda }^n=1$

By [L1], the field $\mathbb{C}$ satisfies the scalar-endomorphism condition of [L2], so it is a splitting field for the finite group $G$. Since $G$ is abelian, [L2] makes every irreducible complex representation of $G$ one-dimensional.

By [L3], every irreducible representation of $G$ is equivalent to a normalized degree-one representation on $\mathbb{C}$, hence to a homomorphism $\chi :G\to {\mathbb{C}}^{\times }$. Because $G=⟨g⟩$, such a homomorphism is determined by the value $\lambda =\chi (g)$, and the relation $g^n=e$ forces ${\lambda }^n=1$. Conversely, if ${\lambda }^n=1$, then ${\chi }_{\lambda }(g^m):={\lambda }^m$ is well defined and multiplicative.

By [L4], the polynomial $t^n-1$ splits over $\mathbb{C}$ and has exactly $n$ distinct roots there, namely the elements of ${\mu }_n(\mathbb{C})$. So step 2.1 produces exactly $n$ equivalence classes of irreducible complex representations, represented by the normalized characters ${\chi }_{\lambda }$ with $\lambda \in {\mu }_n(\mathbb{C})$.