# Quantalic fields - SS 2018

## Lecture and Seminar:

Tuesday 9:45 - 11:15 Uhr
Thursday 9.45 - 11.15 Uhr
Begin: Tuesday, 13 February 2018
Seminar-Room 7.527

Is our knowledge of the main architecture of our number system really complete - apart from specific or notoriously difficult problems yet to be solved? Or is the number system just a trivial part of mathematics - not worth of a reinspection? Gauss, Riemann, and Hilbert did not think so. They found a lot of mysteries therein. Dedekind and Weber (Theory of algebraic functions of a single variable, Crelle~92, 1882) wrote a long paper on function fields (Part I) and Riemann surfaces (Part II), a predecessor of Hilbert's 1897 Zahlbericht. An English version appeared in 1998 - Hilbert (1862-1943) is still alive.

During the whole 20th century, the mysterious ties between numbers (number fields) and functions (living on a Riemann surface), have not stopped to tantalize mathematicians in their dreams, those who are hunting for the "field of constants" in a number field. Several solutions have been offered, but the "philosopher's stone" has not been found.

Ordered structures, after an intensive study in the sixties of the past century, now still waiting for a renaissance, might indicate a way out. The observation that quantales are deeply involved in the arithmetic of field extensions, has been a starting point of our lecture. Notes on quantalic field theory are here.

## Problems:

1. Is there a weak version of modularity which holds in every q-field?

2. Study the category of q-fields (q-groups) - projective/injective objects, generators, zero element, kernels, cokernels ...

3. Is a one-dimensional modular q-field with V(F) strongly independent connected? (cf. Theorem 1). (Prove the strong approximation theorem for q-fields first, using Corollary 2 of Theorem 3.)

4. Study q-fields as q-groups, rational orders as subgroups, prime orders as (positive cones of) linear preorders, etc.

5. Is the double D(F) of a function q-field F generated by K and JF ?

6. Consider a "complete double" instead of D(F) where [0,K] is a q-field.

7. What is the role of the "birational" group G (Cor. 3 of Prop. 29) in the completed double?