= Solution
Choose a positive <root system> $R^+$ for a complex <semisimple Lie algebra>, with <Weyl group> $W$ and <Weyl vector> $\rho=\frac12\sum_{\alpha\in R^+}\alpha$. For a <dominant integral weight> $\lambda$, let $L(\lambda)$ denote its finite-dimensional <irreducible representation>, and use formal symbols with $e^\mu e^\nu=e^{\mu+\nu}$. Its <formal character of a weight module> is $\operatorname{ch}L(\lambda)=\sum_\mu\dim L(\lambda)_\mu\,e^\mu$. The <Weyl character formula> says
$$
\boxed{\operatorname{ch}L(\lambda)
=\frac{\sum_{w\in W}\det(w)e^{w(\lambda+\rho)}}
{\sum_{w\in W}\det(w)e^{w\rho}}.}
$$
Here $\det(w)=(-1)^{\ell(w)}$, the sign of the product of simple <root reflections>. The <Weyl denominator formula> is
$$
\boxed{\sum_{w\in W}\det(w)e^{w\rho}
=e^\rho\prod_{\alpha\in R^+}(1-e^{-\alpha}).}
$$
The quotient is initially interpreted in the fraction field of the group algebra of the <weight lattice>; the theorem makes it a finite character.
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