Title: Ring-Theoretic Properties of Analytic Functions over the Complex Numbers with the p-adic Metric
Speaker: Nick Bruno (Ohio State University)
Abstract: The Complex numbers are the smallest algebraically and metrically closed field over the rational numbers with the traditional Euclidean metric. Using the p-adic metric for some prime $p$ on the rational numbers, we may construct a smallest algebraically and metrically closed field $C_p$ which is isomorphic to the complex numbers. We then consider the rings $A_r$ of analytic functions on $C_p$, specifically convergent power series on some open disk with (possibly infinite) radius $r$. Taking inspiration from the work of O. Helmer and M. Henrickson, I will show that $A_r$ is Bezout if and only if $r$ is infinite, if $r$ is finite every finitely generated ideal has at most two generators, and elements of Spec$(A_r)$ are in one-to-one correspondence with ultrafilters on zero sets.
