[proofplan]
We compute $|G|/\chi(1)$ as a sum that is manifestly an algebraic integer. Starting from the column-orthogonality value $\langle \chi, \chi \rangle = 1$, we rewrite the sum over $G$ as a sum over conjugacy classes and divide by $\chi(1)$. Each summand factors as $\omega_\chi(\mathcal{C}_i) \cdot \chi(g_i^{-1})$, where $\omega_\chi(\mathcal{C}_i) = |\mathcal{C}_i|\chi(g_i)/\chi(1)$ is an algebraic integer by the [Central Character Algebraic Integrality](/theorems/2463), and $\chi(g_i^{-1})$ is an algebraic integer by [Character Values are Algebraic Integers](/theorems/2461). Since algebraic integers form a ring, $|G|/\chi(1)$ is an algebraic integer. It is also a positive rational, so it must be an ordinary positive integer, giving $\chi(1) \mid |G|$.
[/proofplan]
[step:Expand $\langle \chi, \chi \rangle = 1$ using row orthogonality and group the sum by conjugacy classes]
Since $\chi$ is irreducible, the [Row Orthogonality Relations](/theorems/2430) give
\begin{align*}
\langle \chi, \chi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g)\, \overline{\chi(g)} = 1,
\end{align*}
hence
\begin{align*}
|G| = \sum_{g \in G} \chi(g)\, \overline{\chi(g)}.
\end{align*}
By the [Elementary Properties of Characters](/theorems/2421), $\overline{\chi(g)} = \chi(g^{-1})$ for every $g \in G$. Substituting,
\begin{align*}
|G| = \sum_{g \in G} \chi(g)\, \chi(g^{-1}).
\end{align*}
Let $\mathcal{C}_1, \ldots, \mathcal{C}_k$ be the conjugacy classes of $G$ with representatives $g_1, \ldots, g_k$. The function $g \mapsto \chi(g)\chi(g^{-1})$ is constant on each conjugacy class (as $\chi$ is a class function and $g \mapsto g^{-1}$ permutes elements within each class). Therefore the sum over $G$ collapses to a sum over conjugacy classes:
\begin{align*}
|G| = \sum_{i=1}^k |\mathcal{C}_i|\, \chi(g_i)\, \chi(g_i^{-1}).
\end{align*}
[/step]
[step:Divide by $\chi(1)$ to expose $\omega_\chi(\mathcal{C}_i)$ in each summand]
Since $\chi(1) = \dim V \neq 0$, we may divide both sides by $\chi(1)$:
\begin{align*}
\frac{|G|}{\chi(1)} = \sum_{i=1}^k \frac{|\mathcal{C}_i|\, \chi(g_i)}{\chi(1)} \cdot \chi(g_i^{-1}) = \sum_{i=1}^k \omega_\chi(\mathcal{C}_i)\, \chi(g_i^{-1}),
\end{align*}
where we have used the formula
\begin{align*}
\omega_\chi(\mathcal{C}_i) = \frac{|\mathcal{C}_i|\, \chi(g_i)}{\chi(1)}
\end{align*}
for the central character on the $i$-th class sum, established in [Central Character Algebraic Integrality](/theorems/2463).
[/step]
[step:Identify the right-hand side as an algebraic integer]
We verify each summand $\omega_\chi(\mathcal{C}_i)\, \chi(g_i^{-1})$ is an algebraic integer.
First, $\omega_\chi(\mathcal{C}_i)$ is an algebraic integer by [Central Character Algebraic Integrality](/theorems/2463). The hypotheses required there — namely that $G$ is finite and $\chi$ is irreducible — are exactly the hypotheses of the present theorem.
Second, $\chi(g_i^{-1})$ is an algebraic integer by [Character Values are Algebraic Integers](/theorems/2461). The hypothesis required there — that $G$ is finite and $\chi$ is a character of a finite-dimensional representation — is satisfied since $|G| < \infty$ and $\chi$ is the character of a finite-dimensional irreducible representation.
The set of algebraic integers in $\mathbb{C}$ is a subring of $\mathbb{C}$: it is closed under both addition and multiplication. Hence each product $\omega_\chi(\mathcal{C}_i)\, \chi(g_i^{-1})$ is an algebraic integer, and the finite sum
\begin{align*}
\frac{|G|}{\chi(1)} = \sum_{i=1}^k \omega_\chi(\mathcal{C}_i)\, \chi(g_i^{-1})
\end{align*}
is an algebraic integer.
[guided]
We need both $\omega_\chi(\mathcal{C}_i)$ and $\chi(g_i^{-1})$ to be algebraic integers, and we need the algebraic integers to be closed under the ring operations $+$ and $\cdot$.
The first input — that $\omega_\chi(\mathcal{C}_i) = |\mathcal{C}_i|\chi(g_i)/\chi(1)$ is an algebraic integer — is the deep step. It is **not** obvious from the formula alone, since $\chi(1)$ appears in the denominator. The reason is structural: the central character $\omega_\chi: Z(\mathbb{C}G) \to \mathbb{C}$ sends the class sums $C_1, \ldots, C_k$ (which form a $\mathbb{Z}_{\geq 0}$-integer-structure-constant basis for $Z(\mathbb{C}G)$) into a finitely generated $\mathbb{Z}$-submodule of $\mathbb{C}$ that contains $1$ and is invariant under multiplication by each $\omega_\chi(\mathcal{C}_i)$. That is the content of [Central Character Algebraic Integrality](/theorems/2463), and it is exactly the standard $\mathbb{Z}$-module characterisation of algebraic integers.
The second input — that $\chi(g_i^{-1})$ is an algebraic integer — comes from the spectral picture: $\chi(g_i^{-1}) = \operatorname{tr}\rho(g_i^{-1})$ is a sum of eigenvalues of $\rho(g_i^{-1})$, and each eigenvalue is a root of unity (since $\rho(g_i^{-1})$ has finite order in $\operatorname{GL}(V)$, as $g_i^{-1}$ has finite order in $G$). Each root of unity is an algebraic integer (root of $X^n - 1 \in \mathbb{Z}[X]$), and sums of algebraic integers are algebraic integers.
Closure under $+$ and $\cdot$ is the standard fact that algebraic integers form a subring of $\mathbb{C}$, proved using the same finitely-generated-$\mathbb{Z}$-module characterisation: if $\alpha M \subseteq M$ and $\beta N \subseteq N$ for finitely generated $M, N$, then $(\alpha\beta)(MN) \subseteq MN$ and $(\alpha + \beta)(MN) \subseteq MN$, where $MN$ is the $\mathbb{Z}$-module generated by all products $mn$ ($m \in M$, $n \in N$), still finitely generated.
[/guided]
[/step]
[step:Conclude $|G|/\chi(1)$ is a positive integer]
The number $|G|/\chi(1)$ is a positive rational number — both $|G|$ and $\chi(1)$ are positive integers. We have just shown it is also an algebraic integer.
We use the standard fact that any rational number that is also an algebraic integer is an ordinary integer in $\mathbb{Z}$. (Proof: if $\alpha = p/q \in \mathbb{Q}$ in lowest terms satisfies a monic integer polynomial $\alpha^n + c_{n-1}\alpha^{n-1} + \cdots + c_0 = 0$, multiplying by $q^n$ gives $p^n + c_{n-1}p^{n-1}q + \cdots + c_0 q^n = 0$, so $q \mid p^n$. Since $\gcd(p, q) = 1$, this forces $q = \pm 1$, hence $\alpha \in \mathbb{Z}$.)
Applying this to our positive rational $|G|/\chi(1)$, we obtain $|G|/\chi(1) \in \mathbb{Z}$, i.e.\ $\chi(1) \mid |G|$.
[/step]
