In addition to the standard topics, this volume contains many topics not often found in an algebra book, such as inequalities, and the elements of substitution theory. Especially extensive is Chrystal's treatment of the infinite series, infinite products, and (finite and infinite) continued fractions.

Contains historical background and discussion of results for each chapter, References, and an Index.

This volume presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces.

In this second English edition of Caratheodory's work (originally published in German), the two volumes of the first edition have been combined into one (with a combination of the two indexes into a single index). There is a deep and fundamental relationship between the differential equations that occur in the calculus of variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This basic fact forms the rationale for Caratheodory's work.

This second volume is a comprehensive treatment of Diophantine analysis. Besides the familiar cases of Diophantine equations, this rubric also covers partitions, representations as a sum of two, three, four or $n$ squares, Waring's problem in general and Hilbert's solution of it, and perfect squares in artihmetical and geometrical progressions. Many important Diophantine equations, such as Pell's equation, and classes of equations, such as quadratic, cubic and quartic equations, are treated in detail.

Covers the related topics of divisibility and primality. This book describes the development of our understanding of perfect numbers. It also covers standard topics, such as Fermat's theorems, primitive roots, counting divisors, the Mobius function, and prime numbers themselves.

Presents a general theory of iteration algorithms for the numerical solution of equations and systems of equations. This book investigates the relationship between the quantity and the quality of information that is used by an algorithm.

Ramanujan occupies a unique place in analytic number theory: his formulas, identities and calculations are still amazing mathematicians three-quarters of a century after his death. Many of his discoveries seem to have appeared as if from the ether. His mentor and primary collaborator was G.H. Hardy. Here, Hardy collects 12 of his own lectures on topics stemming from Ramanujan's life and work. The topics include: partitions, hypergeometric series, Ramanujan's tau-function and round numbers.

David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This text is his treatment of symbolic logic which lays the groundwork for his later work with Bernays. This translation is based on the second German edition, and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Godel's completeness proof for the predicate calculus has been updated."

Contains many topics in algebra such as inequalities, and the elements of substitution theory. This volume includes over 2,400 exercises with solutions. It also gives a treatment of the infinite series, infinite products, and (finite and infinite) continued fractions.