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?