Axiomatizing denseness in real and p-adic closures
- Date: Wednesday 4 December 2019, 16:00 – 17:00
- Location: Mathematics Level 8, MALL 1, School of Mathematics
- Type: Logic, Seminars, Pure Mathematics
- Cost: Free
Sylvy Anscombe, University of Central Lancashire. Part of the Logic Seminar Series.
Each algebraic field extension of the rational numbers is dense in its all of its real and p-adic closures. Motivated by that simple fact, we study this 'denseness' property and examine when it is elementary in the language of rings. It is not always elementary, but for fields with finite Pythagoras/p-Pythagoras numbers, it is so. In particular, it is elementary modulo the common theory T_Alg of algebraic extensions of Q, and so it is also entailed by T_Alg. We contrast this with the variety of models of T_Alg, ranging from elementary extensions of Q to PAC, PRC, and PpC fields. This is joint work with Philip Dittmann and Arno Fehm.