Prime factorization, class groups, and generalizations
Jim Brown
Department of Mathematical Sciences
Clemson University
We all learn early in our study of mathematics that each
integer has a unique (up to order) factorization into
primes. It turns out that for some prominent
problems arising in number theory it is vital to know if
one can uniquely factor elements of certain nice rings
containing the integers into primes. (Spoiler: it
isn't always possible!) Once one knows there are
rings R for which it isn't possible to factor elements
uniquely into primes, natural questions are can we
determine which R do not have unique factorization and for
those that don't, can we measure how far they are from
having unique factorization? This talk will explore these
questions for rings mathbb{Z}[\zeta_{n}]where zeta_{n} =
e^{2 \pi i /n}as well as generalizations of these
problems.