Oberseminar Algebra und Geometrie

Florian Pop (University of Pennsylvania, Philadelphia)

Abstract:
Recall that the Hilbert Problem 10 (HP10) has a negative answer (by work of Davis, Putman, Julia Robinson, culminating with Matijasevich). That implies at least intuitively that the arithmetic of global fields is “very complicated.” An intriguing question arising from the negative solution to HP10 is whether the arithmetic is so complicated that the isomorphism type of every global field can be “encoded” in a single field axiom, i.e., whether for every global field $K$ there is a field axiom, say $\varphi_K$, such that for all global fields $L$ one has $L\cong K$, provided $\varphi_K$ is true in $L$. It was shown by Rumely in 1980 that such a field axiom $\varphi_K$ does exist indeed for every global field $K$. In my talk I will explain all the terms, and show that the above fact is true for  all finitely generated fields. This is joint work with Philip Dittmann.