Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E. Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function elds. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.

This paper is the first part of a larger project which will give a comprehensive view of the history around the Riemann hypothesis for function fields. (A preliminary version has appeared 1997/98.) This Part 1 is dealing with the development before Hasse's contributions to the Riemann hypothesis. We are trying to explain what he could build upon. The time

One of the fundamental questions in current field theory, related to Grothendieck’s conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the fields themselves. In this paper we initiate the classification of additive properties of multiplicative subgroups of fields containing all squares, using pro-2-Galois groups of nilpotency class at most 2, and of exponent at most 4. This work extends some powerful methods and techniques from formally real fields to general fields of characteristic not 2.

The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kursch'ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of p-adic fields as defined by Kurt Hensel. In the following decades we can observe a quick development of valuation theory, triggered mainly by the discovery that much of algebraic number theory could be better understood by using valuation theoretic notions and methods. An outstanding figure in this development was Helmut Hasse. Independent of the application to number theory, there were essential contributions to valuation theory given by Alexander Ostrowski, published 1934. About the same time Wolfgang Krull gave a more general, universal definition of valuation which turned out to be applicable also in many other mathematical disciplines such as algebraic geometry or functional analysis, thus opening a new era of valuation theory.

i matematik som framläggs för offentlig granskning

Abstract not found

Abstract not found

Communicated by xxx Let

The algebraic closure of R is C, which is a finite extension. Are there other fields which are not algebraically closed but have an algebraic closure which is a finite extension? Yes. An example is the field of real algebraic numbers. Since complex conjugation is a field automorphism fixing Q, and the real and imaginary parts of a complex number can be