Advanced Modern Algebra

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 generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic $K$-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization. (GSM/114)

[1]  W. Burnside,et al.  Theory of Groups of Finite Order , 1909 .

[2]  Florian Cajori,et al.  A history of mathematical notations , 1928 .

[3]  G. Hardy,et al.  An Introduction to the Theory of Numbers , 1938 .

[4]  H. Weyl.  The Classical Groups , 1939 .

[5]  R. Thrall,et al.  Rings with minimum condition , 1944 .

[6]  J. Cassels,et al.  The Theory of Algebraic Numbers (Carus Mathematical Monograph Number Nine). By Harry Pollard. Pp. 143. 24s. 1950. (John Wiley, New York : Chapman and Hall) , 1951, The Mathematical Gazette.

[7]  Jean-Pierre Serre,et al.  Algèbre locale : multiplicités , 1957 .

[8]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[9]  C. Weibel,et al.  AN INTRODUCTION TO HOMOLOGICAL ALGEBRA , 1996 .

[10]  I. Niven,et al.  An introduction to the theory of numbers , 1961 .

[11]  O. O’Meara.  Introduction to quadratic forms , 1965 .

[12]  Walter Feit,et al.  Characters of finite groups , 1965 .

[13]  George W. Polites,et al.  An introduction to the theory of groups , 1968 .

[14]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[15]  Joaquín Pascual,et al.  Infinite Abelian Groups , 1970 .

[16]  John Milnor,et al.  Introduction to algebraic K-theory , 1971 .

[17]  Pierre Samuel,et al.  Algebraic theory of numbers , 1971 .

[18]  S. Lane.  Categories for the Working Mathematician , 1971 .

[19]  I. G. MacDonald,et al.  Lectures on Lie Groups and Lie Algebras: Simple groups of Lie type , 1995 .

[20]  Tsit Yuen Lam,et al.  The algebraic theory of quadratic forms , 1973 .

[21]  C. Procesi.  Rings with polynomial identities , 1973 .

[22]  J. Humphreys.  Introduction to Lie Algebras and Representation Theory , 1973 .

[23]  L. Gaal.  Classical Galois Theory, With Examples , 1973 .

[24]  C. Small,et al.  The Brauer Group of Commutative Rings , 1975 .

[25]  I. Isaacs.  Character Theory of Finite Groups , 1976 .

[26]  T. S. Blyth.  Module Theory: An Approach to Linear Algebra , 1977 .

[27]  P. J. Higgins.  FIELD THEORY AND ITS CLASSICAL PROBLEMS (The Carus Mathematical Monographs, number 19) , 1980 .

[28]  Louis Rowen,et al.  Polynomial identities in ring theory , 1980 .

[29]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[30]  John McCleary,et al.  User's Guide to Spectral Sequences , 1985 .

[31]  H. S. M. Coxeter,et al.  Geometry and Algebra in Ancient Civilizations , 1985 .

[32]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[33]  M. Reid,et al.  Commutative ring theory: Regular rings , 1987 .

[34]  D. Mumford.  The red book of varieties and schemes , 1988 .

[35]  Jean-Pierre Tignol,et al.  Galois' theory of algebraic equations , 1988 .

[36]  J. Stillwell.  Mathematics and Its History , 2020, Undergraduate Texts in Mathematics.

[37]  Leonard Evens,et al.  Cohomology of groups , 1991, Oxford mathematical monographs.

[38]  Alexander J. Hahn,et al.  Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups , 1993 .

[39]  W. Bruns,et al.  Cohen-Macaulay rings , 1993 .

[40]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: Preface , 1994 .

[41]  Y. Drozd,et al.  Finite dimensional algebras , 1994 .

[42]  D. Gorenstein,et al.  The Classification of the Finite Simple Groups , 1983 .

[43]  R. James Milgram,et al.  Cohomology of finite groups , 1994 .

[44]  Charles C. Sims,et al.  Computation with finitely presented groups , 1994, Encyclopedia of mathematics and its applications.

[45]  J. Rosenberg,et al.  Algebraic K-Theory and Its Applications , 1995 .

[46]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[47]  J. Rotman.  A First Course in Abstract Algebra , 1995 .

[48]  Igor R. Shafarevich,et al.  Algebra IX - finite groups of Lie type ; finite-dimensional division algebras , 1996, Encyclopaedia of mathematical sciences.

[49]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[50]  J. Neukirch,et al.  Cohomology of number fields , 2000 .

[51]  M. Aschbacher,et al.  Finite Group Theory: p -groups , 2000 .

[52]  J. McConnell,et al.  Noncommutative Noetherian Rings , 2001 .