Series: Universitext
2006, Approx. 200 p., Softcover
ISBN: 0-387-28725-6
About this textbook
Classical Galois theory is a subject generally acknowledged to be
one of the most central and beautiful areas in pure mathematics.
This text develops the subject systematically and from the
beginning, requiring of the reader only basic facts about
polynomials and a good knowledge of linear algebra.
Key topics and features of this book:
* Approaches Galois theory from the linear algebra point of view,
following Artin.
* Develops the basic concepts and theorems of Galois theory,
including algebraic, normal, separable, and Galois extensions,
and the Fundamental Theorem of Galois Theory.
* Presents a number of applications of Galois theory, including
symmetric functions, finite fields, cyclotomic fields, algebraic
number fields, solvability of equations by radicals, and the
impossibility of solution of the three geometric problems of
Greek antiquity.
* Excellent motivaton and examples throughout
The book discusses Galois theory in considerable generality,
treating fields of characteristic zero and of positive
characteristic with consideration of both separable and
inseparable extensions, but with a particular emphasis on
algebraic extensions of the field of rational numbers. While most
of the book is concerned with finite extensions, it concludes
with a discussion of the algebraic closure and of infinite Galois
extensions.
Steven H. Weintraub is Professor and Chair of the Department of
Mathematics at Lehigh University. This book, his fifth, grew out
of a graduate course he taught at Lehigh. His other books include
Algebra: An Approach via Module Theory (with W. A. Adkins).
Table of contents
Introduction to Galois Theory.- Field Theory and Galois Theory.-
Development and Applications of Galois Theory.- Extensions of the
Field of Rational Numbers.- Further Topics in Field Theory.- A.
Some Results from Group Theory.- B. A Lemma on Constructing
Fields.- C. A Lemma from Elementary Number Theory.- References.-
Index.
Series: Universitext
2006, Approx. 300 p. 6 illus., Softcover
ISBN: 0-387-28930-5
About this textbook
From Math Reviews: "This is a charming textbook, introducing
the reader to the classical parts of algebra. The exposition is
admirably clear and lucidly written with only minimal
prerequisites from linear algebra. The new concepts are, at least
in the first part of the book, defined in the framework of the
development of carefully selected problems. Thus, for instance,
the transformation of the classical geometrical problems on
constructions with ruler and compass in their algebraic setting
in the first chapter introduces the reader spontaneously to such
fundamental algebraic notions as field extension, the degree of
an extension, etc... The book ends with an appendix containing
exercises and notes on the previous parts of the book. However,
brief historical comments and suggestions for further reading are
also scattered through the text."
Table of contents
Foreword.- Constructibility with Ruler and Compass.- Algebraic
Extensions.- Simple Extensions.- Fundamentals of Divisibility.-
Prime Factorization in Polynomial Rings. Gauss's Theorem.-
Polynomial Splittin Fields.- Separable Extensions.- Galois
Extensions.- Finite fields, Cyclic Groups and Roots of Unity.-
Group Actions.- Applications of Galois Theory to Cyclotomic
Fields.- Further Steps into Galois theory.- Norm and Trace.-
Binomial Equations.- Solvability of Equations.- Integral Ring
Extensions.- The Transcendence of pi.- Transcendental Field
Extensions.- Hilbert's Nullstellensatz.- Appendix: Problems and
Remarks.- Index of Notation.- Index
Series: Undergraduate Texts in Mathematics
2005, Approx. 285 p. 48 illus., Softcover
ISBN: 0-387-28723-X
About this textbook
The axiomatic theory of sets is a vibrant part of pure
mathematics, with its own basic notions, fundamental results, and
deep open problems. At the same time, it is often viewed as a
foundation of mathematics so that in the most prevalent, current
mathematical practice "to make a notion precise" simply
means "to define it in set theory." This book tries to
do justice to both aspects of the subject: it gives a solid
introduction to "pure set theory" through transfinite
recursion and the construction of the cumulative hierarchy of
sets (including the basic results that have applications to
computer science), but it also attempts to explain precisely how
mathematical objects can be faithfully modeled within the
universe of sets. In this new edition the author added solutions
to the exercises, and rearranged and reworked the text in several
places to improve the presentation.
The book is aimed at advanced undergraduate or beginning graduate
mathematics students and at mathematically minded graduate
students of computer science and philosophy.
Table of contents
Introduction * Equinumerosity * Paradoxes and axioms * Are sets
all there is? * The natural numbers * Fixed points * Well ordered
sets * Choices * Choice’s consequences * Baire space *
Replacement and other axioms * Ordinal numbers * A. The real
numbers * B. Axioms and universes * Index
Series: Universitext
2006, Approx. 260 p. 5 illus., Hardcover
ISBN: 0-387-28720-5
About this textbook
This book is an accessible introduction to stochastic integration
for students with background in advanced calculus and elementary
probability theory. Further, the author incorporates methods from
measure theory as well as a bit of elementary Hilbert space
theory as applied to L^2 spaces. There are clear examples used to
motivate the concepts and to illustrate the theorems, along with
many exercises at the end of each chapter. Topics include
constructions of Brownian motion, the Ito formula, stochastic
integrals for martingales, and stochastic differential equations.
This text is based on lectures first given in 1998, and has been
used since then for courses taught by the author at Louisiana
State University.
Table of contents
* Introduction * Brownian motion * Constructions of Brownian
motion * Stochastic integrals * An extentions of stochastic
integrals * Stochastic integrals for martingales * The Ito
formula * Multiple Wiener integrals * Stochastic differential
equations * Applications to finance * References
Series: Springer Monographs in Mathematics
2006, Approx. 400 p., Hardcover
ISBN: 3-540-28380-3
About this book
Singularity theory is a young but rapidly growing area of
mathematics with fascinating relations to algebraic geometry,
complex analysis, commutative algebra, representation theory,
theory of Lie groups, topology, dynamical systems, and many more,
and with numerous applications in the natural and technical
sciences.
This book presents the basic singularity theory of analytic
spaces, including local deformation theory, and the theory of
plane curve singularities. Plane curve singularities are on the
one hand easier and on the other hand richer than general
singularities. They are classical but still in the center of
current research and provide an ideal introduction to the general
theory. Deformation theory is one of the most important
techniques in many branches of contemporary mathematics and this
introductory text provides the general framework while still
remaining concrete and applying it to hypersurface singularities.
In the first part of the book the authors develop the relevant
techniques, including Weierstras preparation theorem, the finite
coherence theorem etc., and then locally treat isolated
hypersurface and plane curve singularities, including the finite
determinacy, classification of simple singularities, topological
and analytic invariants, resolution. In the local deformation
theory emphasis is put on the issues of the versality,
obstructions, equisingular deformations. The book contains
moreover a full and new treatment of equisingular deformations of
plane curve singularities including a proof for the smoothness of
the mu-constant stratum which is based on deformations of the
parametrization. The material, which can partly be found in other
books and partly in research articles, is for the first time
exposed from a unified point of view, is supplied by complete
proofs (new in many cases), and can serve as source for special
courses in singularity theory. Computational aspects of the
theory are discussed as well. Three appendices, including basic
facts from the sheaf theory, commutative algebra, and formal
deformation theory, make the reading self-contained.
Table of contents
I. Singularity Theory.- Basic Properties of Complex Spaces and
Germs.- Weierstrass Preparation and Finiteness Theorem.-
Application to Analytic Algebras.- Complex Spaces.- Complex Space
Germs and Singularities.- Finite Morphisms and Finite Coherence
Theorem.- Applications of the Finite Coherence Theorem.- Finite
Morphisms and Flatness.- Flat Morphisms and Fibres.- Singular
Locus and Differential Forms.- Hypersurface Singularities.-
Invariants of Hypersurface Singularities.- Finite Determinacy.-
Algebraic Group Actions.- Classification of Simple Singularities.-
Plane Curve Singularities.- Parametrization.- Intersection
Multiplicity.- Resolution of Plane Curve Singularities.-
Classical Topological and Analytic Invariants
II. Local Deformation Theory.- Deformations of Complex Space
Germs.- Deformations of Singularities.- Embedded Deformations.-
Versal Deformations.- Infinitesimal Deformations.- Obstructions.-
Equisingular Deformations of Plane Curve Singularities.-
Equisingular Deformations of the Equation.- The Equisingularity
Ideal.- Deformations of the Parametrization.- Computation of T^1
and T^2 .- Equisingular Deformations of the Parametrization.-
Equinormalizable Deformations.- Versal Equisingular Deformations.-Appendices:
Sheaves.- Commutative Algebra.- Formal Deformation Theory.-
Literature.- Index
Series: Undergraduate Texts in Mathematics
2006, XII, 250 p. 40 illus., Hardcover
ISBN: 0-387-29139-3
About this textbook
This much-anticipated textbook illuminates the field of discrete
mathematics with examples, theory, and applications of the
discrete volume of a polytope. The authors have weaved a unifying
thread through basic yet deep ideas in discrete geometry,
combinatorics, and number theory. Because there is no other book
that puts together all of these ideas in one place, this text is
truly a service to the mathematical community.
We encounter here a friendly invitation to the field of "counting
integer points in polytopes", also known as Ehrhart theory,
and its various connections to elementary finite Fourier
analysis, generating functions, the Frobenius coin-exchange
problem, solid angles, magic squares, Dedekind sums,
computational geometry, and more. With 250 exercises and open
problems, the reader will feel like an active participant, and
the authors' engaging style will encourage such participation.
For teachers, this text is ideally suited as a capstone course
for undergraduate students or as a compelling text in discrete
mathematical topics for beginning graduate students. For
scientists, this text can be utilized as a quick tooling device,
especially for those who want a self-contained, easy-to-read
introduction to these topics.
Table of contents
* Introduction * The coin-exchange problem of Frobenius * Magic
Squares * The Ehrhart theory * Reciprocity * Face numbers and the
Dehn-Sommerville relations * Finite Fourier analysis * Dedekind
sums * Complex-analytic methods * The decomposition of a polytope
into its cones * Euler-MacLaurin summation in Rd * A connection
to elliptic functions * References * Index