Factorial invariant ring with no linear characters
= Factorial invariant ring with no linear characters
If a <finite group> has no nontrivial homomorphism to $\mathbb C^\times$, its <polynomial invariant ring> is a <unique factorization domain>. Factor an invariant in the ambient <polynomial ring> and collect its irreducible factors into orbit products. Each orbit product transforms by a <linear character>, hence is invariant. It is prime in the invariant ring, and the invariant factorization is a product of these primes.