Advanced Modern Algebra

  title={Advanced Modern Algebra},
  author={Joseph J. Rotman},
This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely… 
A primer on computational group homology and cohomology
These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra
Exploring the Beginnings of Algebraic K-Theory
According to Atiyah, K-theory is that part of linear algebra that studies additive or abelian properties (e.g. the determinant). Because linear algebra, and its extensions to linear analysis, is
Alternating Forms and the Brauer Group of a Field Containing the Roots of Unity
Let F be a field of characteristic 0 that contains μ, the group of all roots of unity. Our aim is to establish a behavioral ideal for the division algebras and Brauer group Br(F ) of F . This ideal
Title of thesis: CLASSIFICATION OF PRIME IDEALS IN INTEGRAL GROUP ALGEBRAS OF FINITE ABELIAN GROUPS Heather Mallie McDonough, Master of Arts, 2005 Thesis directed by: Professor William Adams
Effectiveness in Stallings' Proof of Grushko's Theorem
Algebraically this is useful as it implies every finitely generated group admits a unique decomposition as a free product of indecomposable groups. As an immediate consequence we have additivity of
Twisted group ring isomorphism problem and infinite cohomology groups
We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any
On incidence algebras and their representations
We provide a unified approach, via incidence algebras, to the classification of several important types of representations: distributive, thin, with finitely many orbits, or with finitely many
Pythagorean Triples and Inner Products
Linear algebra over rings particularly over Z—is a fascinating topic that straddles course boundaries and can help bridge the present unnecessary gap between introductions to algebra and number
On the homological algebra of clusters, quivers, and triangulations
This thesis is comprised of three parts. Chapter one contains background material detailing some important aspects of category theory and homological algebra. Beginning with abelian categories, we
Representation theory of algebras related to the partition algebra
The main objective of this thesis is to determine the complex generic representation theory of the Juyumaya algebra. We do this by showing that a certain specialization of this algebra is isomorphic


Cohomology of finite groups
The cohomology of groups has, since its beginnings in the 1920s and 1930s, been the stage for significant interaction between algebra and topology and has led to the creation of important new fields
A First Course in Abstract Algebra
1. Number Theory. Induction. Binomial Coefficients. Greatest Common Divisors. The Fundamental Theorem of Arithmetic. Congruences. Dates and Days. 2. Groups I. Functions. Permutations. Groups.
Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups
This volume introduces mathematicians to the large and dynamic area of algebras and forms over commutative rings. The book begins with elementary aspects and progresses gradually in its degree of
Algebraic K-Theory and Its Applications
Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad
Cohen-Macaulay rings
In this chapter we introduce the class of Cohen–Macaulay rings and two subclasses, the regular rings and the complete intersections. The definition of Cohen–Macaulay ring is sufficiently general to
Galois' theory of algebraic equations
Galois' Theory of Algebraic Equations gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth
Theory of Groups of Finite Order
Preface to the second edition Preface to the first edition 1. On permutations 2. The definition of a group 3. On the simpler properties of a group which are independent of its mode of representation
Introduction to algebraic K-theory
Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ? an abelian group K0? or K1? respectively. Professor Milnor sets
Ideals, Varieties, and Algorithms
(here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In
An Introduction to the Theory of Numbers
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,