Just as matter is made up of atoms, the integers are made up of the primes. Just as one could zoom in farther on an atom to find even smaller particles, one can zoom in on the primes and find that they too are made of smaller parts.
A number field is a finite (thus algebraic) field extension of the rational numbers. In each number field lays an integral domain called its ring of integers. Each ring of integers has its very own set of prime ideals. As one climbs up from one number field to a larger one, many of the primes we had before are no longer prime; some of them become products of distinct primes, some of them become powers of a prime, etc... but how often do primes split in any given way?
If the number field is fixed, such questions have answers going back to Dirichlet and Chebotarev. If instead the prime is fixed, the analogous questions have more modern answers due to mathematicians including but not limited to Melanie Wood and Manjul Bhargava. I have focused my research on density questions about the splitting behavior of primes in number fields which depend on the varying prime. In this talk, after a brief crash course in Algebraic Number Theory, I will describe this research.
I will assume some exposure to Galois Theory. I will also assume some level of comfort with hand-waving.