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An especially timely work, the book is an introduction to the theory of p-adic L-functions originated by Kubota and Leopoldt in 1964 as p-adic analogues of the classical L-functions of Dirichlet.Professor Iwasawa reviews the classical results on Dirichlet's L-functions and sketches a proof for some of them. Next he defines generalized Bernoulli numbers and discusses some of their fundamental properties. Continuing, he defines p-adic L-functions, proves their existence and uniqueness, and treats p-adic logarithms and p-adic regulators. He proves a formula of Leopoldt for the values of p-adic L-functions at s=1. The formula was announced in 1964, but a proof has never before been published. Finally, he discusses some applications, especially the strong relationship with cyclotomic fields.

Number theory --- 511.6 --- Algebraic number theory --- L-functions --- Functions, L --- -Number theory --- Algebraic number fields --- Algebraic number theory. --- L-functions. --- 511.6 Algebraic number fields --- -511.6 Algebraic number fields --- Abelian extension. --- Absolute value. --- Algebraic closure. --- Algebraic number field. --- Algebraic number. --- Algebraically closed field. --- Arithmetic function. --- Class field theory. --- Complex number. --- Conjecture. --- Cyclotomic field. --- Dirichlet character. --- Existential quantification. --- Finite group. --- Integer. --- L-function. --- Mellin transform. --- Meromorphic function. --- Multiplicative group. --- P-adic L-function. --- P-adic number. --- Power series. --- Prime number. --- Quadratic field. --- Rational number. --- Real number. --- Root of unity. --- Scientific notation. --- Series (mathematics). --- Special case. --- Subgroup. --- Theorem. --- Topology. --- Nombres, Théorie des

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In this, one of the first books to appear in English on the theory of numbers, the eminent mathematician Hermann Weyl explores fundamental concepts in arithmetic. The book begins with the definitions and properties of algebraic fields, which are relied upon throughout. The theory of divisibility is then discussed, from an axiomatic viewpoint, rather than by the use of ideals. There follows an introduction to p-adic numbers and their uses, which are so important in modern number theory, and the book culminates with an extensive examination of algebraic number fields. Weyl's own modest hope, that the work "will be of some use," has more than been fulfilled, for the book's clarity, succinctness, and importance rank it as a masterpiece of mathematical exposition.

Algebraic number theory. --- Abelian group. --- Absolute value. --- Abstract algebra. --- Addition. --- Additive group. --- Adjunction (field theory). --- Algebra. --- Algebraic equation. --- Algebraic function. --- Algebraic manifold. --- Algebraic number field. --- Algebraic number theory. --- Algebraic number. --- Algebraic operation. --- Algebraic surface. --- Algebraic theory. --- An Introduction to the Theory of Numbers. --- Analytic function. --- Automorphism. --- Axiomatic system. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Class number. --- Coefficient. --- Commutative property. --- Commutative ring. --- Complex number. --- Cyclic group. --- Cyclotomic field. --- Dimension. --- Direct product. --- Dirichlet series. --- Discriminant. --- Divisibility rule. --- Division algebra. --- Divisor. --- Entire function. --- Equation. --- Euler function. --- Existential quantification. --- Finite field. --- Fractional ideal. --- Functional equation. --- Fundamental theorem of algebra. --- Galois group. --- Galois theory. --- Geometry. --- Ground field. --- Hermann Weyl. --- Ideal number. --- Identity matrix. --- Infinite product. --- Integer. --- Irreducibility (mathematics). --- Irreducible polynomial. --- Lattice (group). --- Legendre symbol. --- Linear map. --- Logarithm. --- Mathematics. --- Meromorphic function. --- Modular arithmetic. --- Multiplicative group. --- Natural number. --- Nth root. --- Number theory. --- P-adic number. --- Polynomial. --- Prime factor. --- Prime ideal. --- Prime number theorem. --- Prime number. --- Prime power. --- Principal ideal. --- Quadratic equation. --- Quadratic field. --- Quadratic form. --- Quadratic reciprocity. --- Quadratic residue. --- Real number. --- Reciprocity law. --- Riemann surface. --- Ring (mathematics). --- Ring of integers. --- Root of unity. --- S-plane. --- Scientific notation. --- Sign (mathematics). --- Special case. --- Square number. --- Subgroup. --- Summation. --- Symmetric function. --- Theorem. --- Theoretical physics. --- Theory of equations. --- Theory. --- Variable (mathematics). --- Vector space.