The values of a central character are algebraic integers

Let $C_1,\dots ,C_r$ be the conjugacy classes of $G$, and let $A$ be the $\mathbb{Z}$-subring of $Z(\mathbb{C}[G])$ generated by the class sums ${\widehat{C}}_1,\dots ,{\widehat{C}}_r$. If ${\widehat{C}}_i{\widehat{C}}_j={\sum }_k{n}_{ij}^k{\widehat{C}}_k$, then for any fixed $z\in C_k$, the coefficient $n_{ij}^k$ counts pairs $(x,y)\in C_i\times C_j$ with $xy=z$. Conjugation bijects these pairs for any two choices of $z\in C_k$, so $n_{ij}^k\in \mathbb{Z}$. Hence the additive group of $A$ is contained in the free abelian group on the finitely many class sums, so $A$ is finitely generated as a $\mathbb{Z}$-module.

The ring $A$ is a faithful $A$-module over itself, and step 1.1 makes it finitely generated over $\mathbb{Z}$. Therefore [F2] shows that each generator ${\widehat{C}}_i$ is integral over $\mathbb{Z}$. Then [F3] implies that every element of $A$ is integral over $\mathbb{Z}$.

Let $V$ be an irreducible representation affording $\chi$. By [F1], the action of $A$ on $V$ sends each $a\in A$ to a scalar ${\lambda }_{\chi }(a)\in \mathbb{C}$, and $a\mapsto {\lambda }_{\chi }(a)$ is a ring homomorphism because the action of $A$ is by endomorphisms. If $m(X)\in \mathbb{Z}[X]$ is monic with $m(\widehat{C})=0$, then applying ${\lambda }_{\chi }$ gives the monic equation $m({\omega }_{\chi }(\widehat{C}))=0$. So ${\omega }_{\chi }(\widehat{C})$ is integral over $\mathbb{Z}$, that is, an algebraic integer.