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16
Class Field Theory in Characteristic p, its Origin and Development
 the Proceedings of the International Conference on Class Field Theory (Tokyo
, 2002
"... 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 (1 ..."
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Cited by 15 (5 self)
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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.
The Riemann hypothesis in characteristic p, its origin and development  Part 1. The formation of the zetafunctions of Artin and of F. K. Schmidt
, 2003
"... 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 hypot ..."
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Cited by 7 (2 self)
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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
Additive Structure of Multiplicative Subgroups of Fields and Galois Theory
 DOCUMENTA MATH.
, 2004
"... 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 classificati ..."
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Cited by 6 (4 self)
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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 pro2Galois 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.
History of Valuation Theory  Part I
"... 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 padic fields as defined by Kurt Hensel. In the following decades we ..."
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Cited by 3 (0 self)
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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 padic 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.
On the field intersection problem of solvable quintic generic polynomials, preprint. arXiv:0804.4875v1 [math.NT
"... Abstract. We study a general method of the field intersection problem of generic polynomials over an arbitrary field k via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using multiresolvent polynomials. 1. ..."
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Cited by 2 (2 self)
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Abstract. We study a general method of the field intersection problem of generic polynomials over an arbitrary field k via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using multiresolvent polynomials. 1.
Classically projective groups and pseudo classically closed fields
"... Introduction Let K be a PAC (pseudo algebraically closed) field. Then the absolute Galois group GK of K is projective by Ax [A], i.e., group extensions of GK by a finite groups H \Gamma! GK ! 1 are split: There exists oe : GK ! F such that oe = id GK . On the other hand, Gruenberg [G] showed tha ..."
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Cited by 2 (1 self)
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Introduction Let K be a PAC (pseudo algebraically closed) field. Then the absolute Galois group GK of K is projective by Ax [A], i.e., group extensions of GK by a finite groups H \Gamma! GK ! 1 are split: There exists oe : GK ! F such that oe = id GK . On the other hand, Gruenberg [G] showed that for a profinite projective group G, all group extensions of G by profinite groups H are split. Using Gruenberg's theorem Lubotzky  van den Dries [LvdD] solved the inverse absolute Galois problem for projective groups as follows: 2000 Mathematics Subject Classification. Primary 11, 12, 14; Secondary 11: G, S20, U09; 12: D, E20, F, L; 14: E, G. c fl0000 American Mathematical Society For projective profinte groups G there exist PAC fields K with GK = G. Extending the category of PAC fields, one introduced the PRC (pseudo real closed) fields, the PpC (pseudo padically closed) fields and further, fields regularly closed with respect to finitely many henselisations and orderings, see
Local discriminants, kummerian extensions, and abelian curves
, 711
"... mit den bisherigen Methoden entweder überhaupt nicht gelang, oder doch ..."
THE ARTINSCHREIER THEOREM
"... 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 t ..."
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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
History of Valuation Theory Part I
"... The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kürschák who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic fields as defined by Kurt Hensel. In the following decades we can observ ..."
Abstract
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The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kürschák who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic 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.