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Elliptic functions --- Algebraic functions --- Local fields (Algebra)

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The p-adic numbers, the earliest of local fields, were introduced by Hensel some 70 years ago as a natural tool in algebra number theory. Today the use of this and other local fields pervades much of mathematics, yet these simple and natural concepts, which often provide remarkably easy solutions to complex problems, are not as familiar as they should be. This book, based on postgraduate lectures at Cambridge, is meant to rectify this situation by providing a fairly elementary and self-contained introduction to local fields. After a general introduction, attention centres on the p-adic numbers and their use in number theory. There follow chapters on algebraic number theory, diophantine equations and on the analysis of a p-adic variable. This book will appeal to undergraduates, and even amateurs, interested in number theory, as well as to graduate students.

Local fields (Algebra) --- 511.6 --- 511.6 Algebraic number fields --- Algebraic number fields --- Fields, Local (Algebra) --- Algebraic fields --- Local fields (Algebra).

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Devoted to counterparts of classical structures of mathematical analysis in analysis over local fields of positive characteristic, this book treats positive characteristic phenomena from an analytic viewpoint. Building on the basic objects introduced by L. Carlitz - such as the Carlitz factorials, exponential and logarithm, and the orthonormal system of Carlitz polynomials - the author develops a kind of differential and integral calculi. He also expands on the basics of an analytic theory of (Carlitz's) differential equations, providing a useful foundation for the study of various special functions. The differential calculus is extended to a type of Rota's umbral calculus, and an investigation is made of the corresponding rings of differential operators. A theory of quasi-holonomic modules over these rings, having some common features with holonomic modules in the sense of Bernstein, is also connected to some special functions in the spirit of Zeilberger's theory.

Mathematical analysis. --- Mathematics. --- Math --- Science --- 517.1 Mathematical analysis --- Mathematical analysis --- Local fields (Algebra) --- Fields, Local (Algebra) --- Algebraic fields

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Algebraic fields --- Homology theory --- Corps algébriques --- Homologie --- Corps algébriques --- Group theory. --- Homology theory. --- Groupes, Théorie des. --- Cohomologie. --- Corps locaux (algèbre) --- Local fields (Algebra) --- Algèbre --- Algebra --- Groupes, Théorie des --- Cohomologie --- Algèbre --- Algebra. --- Corps de classe --- Corps et polynomes

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51 <082.1> --- Mathematics--Series --- Local fields (Algebra) --- Galois theory. --- Représentations de groupes --- Corps locaux (Algèbre) --- Représentations de groupes --- Corps locaux (Algèbre) --- Théorie de Galois --- Ordered algebraic structures --- Representations of groups. --- Galois theory --- Representations of groups --- Fields, Local (Algebra) --- Algebraic fields --- Equations, Theory of --- Group theory --- Number theory --- Group representation (Mathematics) --- Groups, Representation theory of