Product Description
This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index. ... Read more

Customer Reviews (3)

Book of Rare Beauty
This is an exceptionally good introduction to the core definitions and theorems of naive set theory.That is to say, it is written in an informal style, yet remains adequate for building deeper knowledge later. I found the treatment somewhat idiosyncratic in that it gives some insights about how Kaplansky views sets and some natural proofs of their properties. Note: this last point suggests the following analogy: you're in college and you take two logic classes just because you love logic so much. However, both are taught by different teachers. One teacher simply goes from the text; the other one hands out a set of lecture notes that is very personal and interesting and adds insight to the textbook. Kaplansky's work is this set of lecture notes. It is also rare to find coverage of metric spaces in a work of its kind, which adds to its appeal. This book is brief, but is not intended to pry beyond the bare essentials -- the written style is ideally suited to this end. A pleasure to read! Highly recommended for a first go.

Extremely helpful!
Kaplansky's Set Theory and Metric Spaces is one of the most helpful math books that I've ever used.Begining with basic set theory and covering such topics as cardinal numbers, countability, the Axiom of Choice, Zorn's Lemma, well ordering, basic properties of metric spaces, completeness, separability, and compactness, this book covered all of the main topics in my set theory class.Kaplansky's explanations are clear and concise, his examples are illuminating, and the problems are challenging without being far too difficult.As a bonus, this book is quite resonably priced.A must for anyone who is taking a first or second set theory class!

Extremely helpful!
Kaplansky's Set Theory and Metric Spaces is one of the most helpful math books that I've ever used.Begining with basic set theory and covering such topics as cardinal numbers, countability, the Axiom of Choice, Zorn's Lemma, well ordering, basic properties of metric spaces, completeness, separability, and compactness.Kaplansky's explanations are clear and concise, his examples are illuminating, and the problems are challenging without being far too difficult.As a bonus, this book is quite resonably priced.A must for anyone who is taking a first or second set theory class! ... Read more

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Customer Reviews (1)

A compact and Accessible Introduction to Infinite Group Theory
Kaplansky explicitly states in the forward that this book is not so much aimed at those wanting to learn about infinite abelian groups, but rather at those wanting to learn about infinite algebra. While this book is certainly a superb introduction to the theory of infinite abelian groups, it does a better job of teaching familiarity with the methods of proof commonly used in more advanced mathematics. As such, the book is extremely accessible, requiring only the absolute basics of group theory. On the other hand, it does require a certain level of "mathematical maturity," the ability to learn with a minimum of motivation and fill in gaps in proofs. For the specialist looking for a reference on infinite abelian groups, I would recommend both volumes of Laszlo Fuchs' Infinite Abelian Groups, Volume 1, Volume 36-I (Pure and Applied Mathematics), which is aimed more at the professional mathematician than the amature. ... Read more

Product Description
This volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, "Lie Algebras," the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, "The Structure of Locally Compact Groups," deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952. ... Read more

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Irving Kaplansky is one of the top mathematicians of this continent of this century. He was the Director from the MSRI 1984-1992. He has made many contributions to many fields of mathematics and they are very valuable. In this volume, 22 of his research papers are collected. Each is accompanied by an updating afterthought, often including references to later papers by other authors. In addition there are eight hitherto unpublished papers. These range over a variety of topics and will be of interest to many readers. ... Read more

Product Description
This book combines in one volume Irving Kaplansky's lecture notes on the theory of fields, ring theory, and homological dimensions of rings and modules. ... Read more

Customer Reviews (1)

Pretty good
This book is an advanced treatment of field theory and Galois theory and is meant for those readers who have a substantial background in graduate algebra. The subject matter used to be thought of as purely mathematical, but due to the influence of the field of cryptography, it now has many applications. I only read part 1 of the book, so my review will be confined to this part.

The author begins the discussion with field extensions. One can view a field L containing another field K as a vector space over K, and the dimension of L (as a vector space) is then called the 'dimension' of L over K. If one considers a subfield K of a field M, and an additional element u in M, then there is a smallest subfield of M containing K and u. Calling this field K(u), u can be either transcendental or algebraic over K. The author then proves some elementary properties of the field K(u), showing the existence of an irreducible polynomial for u over K. This then motivates him to call a field L containing K 'algebraic' over K if every element of L is algebraic over K. Otherwise L is called 'transcendental' over K. The dimension of K(u) over K is called the degree of u over K. Finding the degree of u can be done by finding the irreducible polynomial for u. The author also proves the arithmetic relation between the dimensions of towers of fields, and this allows him to prove the famous results on the impossibility of ruler and compass constructions. For a field L that lies between fields K and M the author studies the 'stability' of L over K, meaning that every automorphism of M/K sends L into itself. The correspondence between stable fields and normal subgroups of the Galois group of M/K is proven. Splitting fields are introduced as devices to obtain fields that are normal over a given field. A criterion for a splitting field that does not involve polynomials is proven, and the author gives tools that deal with fields of non-zero characteristic, these tools motivating the definition of separability. Splitting fields are normal in characteristic 0, but one must add separability for the same to hold in characteristic p. The unsolvability of the quintic is shown via a discussion on radical extensions of fields. For a field K of characteristic 0, and for a field L lying between K and another field M, where M is a radical extension of K, the author proves in detail that the Galois group of L/K will be solvable. Then if one has a polynomial with coefficients in K, then the Galois group of this polynomial is defined to be the Galois group of a splitting field of the polynomial over K. The Galois group of the polynomial is thought of as a group of permutations of the roots of the polynomial. The author then proves that if K has characteristic 0 and L is a radical extension of K which contains a root of the polynomial, then the Galois group of the polynomial over K is solvable.

Those readers involved in cryptography will find a discussion of finite fields in Part 1. The author's goal is to find the finite fields and determine their structure. He first proves that every nth power of a prime number p will yield a field with p^n elements. The author shows that the Galois theory of finite fields is simple by proving that if K is a finite field contained in another finite field L, then L is normal over K and the Galois group of L/K is cyclic.

The author also shows how the Galois group of an equation can be found explicitly for the cubic and quartic equations. He shows first that for the Galois group of a separable irreducible cubic over a field K is either the alternating group A(3) or the symmetric group S(3). If the characteristic of K is not equal to 2, then it is A(3) if and only if the discriminant is a square in K. For a separable irreducible quartic over K, then for the degree over K of the splitting field of the resolvent cubic of this polynomial, the Galois group is S(4) if the degree is 6, A(4) if the degree is 3, V (a particular normal subgroup of S(4)) if the degree is 1, and either the group of order 8 or cyclic of order 4 if the degree is 2.

Also in part 1, the author studies the reducibility of an equation of the form x^n -a over an arbitrary field. He addresses this reducibility by first proving that one only need be concerned for the case where n is a prime power. Then if p is prime, and "a " does not have any pth root in the field K, then if the prime is odd, then the equation is irreducible over K for any n. If p = 2 and the characteristic of K is 2, then the equation is irreducible over K for any n. If p = 2, n is greater than or equal to 2, and the characteristic of K is not 2, then the equation is irreducible over K if and only if -4a is not a fourth power in K. The author also proves the fundamental theorem of algebra using Galois theory. He does this by first showing that if every extension of K has degree divisible by a prime p, then every extension of K has degree a power of p. ... Read more

Customer Reviews (1)

Good but antiquated.
This book does have the advantage of being terse, well-written, and very good problems.You could learn field theory very quickly.

However, it has the severe disadvantage of using antiquated terminology and notation that make it confusing if not detrimental to learning modern commutative ring theory. ... Read more