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Prime ministers --- Olivares, Gaspar de Guzmán,

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Prime ministers --- Palmerston, Henry John Temple, - Viscount, - 1784-1865. --- Great Britain

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Prime ministers --- Biography --- Walpole, Robert, --- Great Britain --- Great Britain --- Politics and government --- Politics and government

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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 volume, the author covers profinite groups and their cohomology, Galois cohomology, and local class field theory, and concludes with a treatment of duality. His objective is to present effectively that body of material upon which all modern research in Diophantine geometry and higher arithmetic is based, and to do so in a manner that emphasizes the many interesting lines of inquiry leading from these foundations.

Group theory --- Finite groups --- Algebraic number theory --- 512.73 --- 512.66 --- Homology theory --- Number theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Groups, Finite --- Modules (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes, Théorie des. --- Group theory. --- Homology theory. --- Finite groups. --- Algebraic number theory. --- Abelian group. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic extension. --- Algebraic geometry. --- Algebraic number field. --- Brauer group. --- Category of abelian groups. --- Category of sets. --- Characterization (mathematics). --- Class field theory. --- Cohomological dimension. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Composition series. --- Computation. --- Connected component (graph theory). --- Coset. --- Cup product. --- Dedekind domain. --- Degeneracy (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Diophantine geometry. --- Discrete group. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Exponential function. --- Family of sets. --- Field extension. --- Finite group. --- Fundamental class. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Galois module. --- Galois theory. --- General topology. --- Geometry. --- Grothendieck topology. --- Group cohomology. --- Group extension. --- Group scheme. --- Hilbert symbol. --- Hopf algebra. --- Ideal (ring theory). --- Inequality (mathematics). --- Injective sheaf. --- Inner automorphism. --- Inverse limit. --- Kummer theory. --- Lie algebra. --- Linear independence. --- Local field. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Module (mathematics). --- Morphism. --- Natural topology. --- Neighbourhood (mathematics). --- Normal extension. --- Normal subgroup. --- Number theory. --- P-adic number. --- P-group. --- Polynomial. --- Pontryagin duality. --- Power series. --- Prime number. --- Principal ideal. --- Profinite group. --- Quadratic reciprocity. --- Quotient group. --- Ring of integers. --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Supernatural number. --- Sylow theorems. --- Tangent space. --- Theorem. --- Topological group. --- Topological property. --- Topological ring. --- Topological space. --- Topology. --- Torsion group. --- Torsion subgroup. --- Transcendence degree. --- Triviality (mathematics). --- Unique factorization domain. --- Variable (mathematics). --- Vector space. --- Groupes, Théorie des --- Nombres, Théorie des