Galois Theory of Fields and Addition and Multiplication Laws
The basic question of interest for this talk is the following. Does the Galois group of a field (with respect to its algebraic closure or separable closure) determine the field? Looking at the example of the field of real and complex numbers, this question might seem uninteresting. On the other hand it is a surprising theorem that the Galois group of the field of rational numbers (with respect to an algebraic closure) does determine the field of rational numbers uniquely (up to isomorphism). In the past two decades S. Mochizuki has provided a fresh insight into this question, notably that Galois groups do sometimes see the addition law and multiplication law of a field separately and that this provides us with a way of changing the addition law without changing the Galois group. I hope to provide a low brow view of this topic (with plenty of concrete examples) and then, if time permits, discuss some recent work in which I was able to provide an algebraic approach to this idea of Mochizuki by showing that in fact there is still a universal addition law in the theory of p-adic fields from which all relevant addition laws arise. A large part of this talk is intended for a wide audience (with some basic knowledge of Galois theory).