Galois Theory

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Askold Khovanskii - One of the best experts on this subject based on the ideXlab platform.

  • Coverings and Galois Theory
    Springer Monographs in Mathematics, 2014
    Co-Authors: Askold Khovanskii
    Abstract:

    This chapter is devoted to the geometry of coverings and its relation to Galois Theory. There is a surprising analogy between the classification of coverings over a connected, locally connected, and locally simply connected topological space and the fundamental theorem of Galois Theory. We state the classification results for coverings so that this analogy becomes evident.

  • Multidimensional Topological Galois Theory
    Springer Monographs in Mathematics, 2014
    Co-Authors: Askold Khovanskii
    Abstract:

    In topological Galois Theory for functions of one variable (see Chap. 5), it is proved that the way the Riemann surface of a function is positioned over the complex line can obstruct the representability of that function by quadratures. This not only explains why many differential equations are not solvable by quadratures, but also gives the strongest known results on their unsolvability.

  • Topological Galois Theory - Topological Galois Theory
    Springer Monographs in Mathematics, 2014
    Co-Authors: Askold Khovanskii
    Abstract:

    This book provides a detailed and largely self-contained description of various classical and new results on solvability and unsolvability of equations in explicit form. In particular, it offers a complete exposition of the relatively new area of topological Galois Theory, initiated by the author. Applications of Galois Theory to solvability of algebraic equations by radicals, basics of Picard–Vessiot Theory, and Liouville's results on the class of functions representable by quadratures are also discussed. A unique feature of this book is that recent results are presented in the same elementary manner as classical Galois Theory, which will make the book useful and interesting to readers with varied backgrounds in mathematics, from undergraduate students to researchers. In this English-language edition, extra material has been added (Appendices A–D), the last two of which were written jointly with Yura Burda

  • Ramified Coverings and Galois Theory
    Galois Theory Coverings and Riemann Surfaces, 2013
    Co-Authors: Askold Khovanskii
    Abstract:

    The third chapter contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface. For such surfaces, the geometry of ramified coverings and Galois Theory are not only analogous but in fact very closely related to each other. This relationship is useful in both directions. On the one hand, Galois Theory and Riemann’s existence theorem allow one to describe the field of functions on a ramified covering over a Riemann surface as a finite algebraic extension of the field of meromorphic functions on the Riemann surface. On the other hand, the geometry of ramified coverings together with Riemann’s existence theorem allows one to give a transparent description of algebraic extensions of the field of meromorphic functions over a Riemann surface.

Nicholas Pippenger - One of the best experts on this subject based on the ideXlab platform.

  • Galois Theory for minors of finite functions
    Discrete Mathematics, 2002
    Co-Authors: Nicholas Pippenger
    Abstract:

    Abstract The motivating example for our work is given by sets of Boolean functions closed under taking minors. A Boolean function f is a minor of a Boolean function g if f is obtained from g by substituting an argument of f , the complement of an argument of f , or a Boolean constant for each argument of g . The Theory of minors has been used to study threshold functions (also known as linearly separable functions) and their generalization to functions of bounded order (where the degree of the separating polynomial is bounded, but may be greater than one). We construct a Galois Theory for sets of Boolean functions closed under taking minors, as well as for a number of generalizations of this situation. In this Galois Theory we take as the dual objects certain pairs of relations that we call “constraints”, and we explicitly determine the closure conditions on sets of constraints.

  • Galois Theory for Minors of Finite Functions
    1998
    Co-Authors: Nicholas Pippenger
    Abstract:

    Galois Theory for Minors of Finite Functions Nicholas Pippenger A Boolean function f is a minor of a Boolean function g if f is obtained from g by substituting an argument of f, the complement of an argument of f, or a Boolean constant for each argument of g. The Theory of minors has been used to study threshold functions (also known as linearly separable functions) and their generalization to functions of bounded order (where the degree of the separating polynomial is bounded, but may be greater than one). We construct a Galois Theory for sets of Boolean functions closed under taking minors, as well as for a number of generalizations of this situation. In this Galois Theory we take as the dual objects certain pairs of relations that we call ``constraints'''', and we explicitly determine the closure conditions on sets of constraints.

Shreeram S. Abhyankar - One of the best experts on this subject based on the ideXlab platform.

  • Galois Theory of special trinomials
    Revista Matemática Iberoamericana, 2003
    Co-Authors: Shreeram S. Abhyankar
    Abstract:

    This is the material which I presented at the 60th birthday conference of my good friend Jose Luis Vicente in Seville in September 2001. It is based on the nine lectures, now called sections, which were given by me at Purdue in Spring 1997. This should provide a good calculational background for the Galois Theory of vectorial ( = additive) polynomials and their iterates.

  • resolution of singularities and modular Galois Theory
    Bulletin of the American Mathematical Society, 2000
    Co-Authors: Shreeram S. Abhyankar
    Abstract:

    I shall sketch a brief history of the desingularization problem from Riemann thru Zariski to Hironaka, including the part I played in it and the work on Galois Theory which this led me to, and how that caused me to search out many group Theory gurus. I shall also formulate several conjectures and suggest numerous thesis problems.

  • Galois Theory on the line in nonzero characteristic
    arXiv: Number Theory, 1992
    Co-Authors: Shreeram S. Abhyankar
    Abstract:

    The author surveys Galois Theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.

Marius Van Der Put - One of the best experts on this subject based on the ideXlab platform.

Michael F. Singer - One of the best experts on this subject based on the ideXlab platform.

  • Galois Theory of linear differential equations
    2012
    Co-Authors: Michael F. Singer
    Abstract:

    Linear differential equations form the central topic of this volume, Galois Theory being the unifying theme. A large number of aspects are presented: algebraic Theory especially differential Galois Theory, formal Theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used. This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

  • Galois Theory of Difference Equations
    Lecture Notes in Mathematics, 1997
    Co-Authors: Michael F. Singer
    Abstract:

    This book lays the algebraic foundations of a Galois Theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois Theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers

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