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.
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This book is an informal and readable introduction to higher algebra at
the post-calculus level. The concepts of ring and field are introduced
through study of the familiar examples of the integers and polynomials.
The new examples and theory are built in a well-motivated fashion and
made relevant by many applications - to cryptography, coding,
integration, history of mathematics, and especially to elementary and
computational number theory. The later chapters include expositions of
Rabiin's probabilistic primality test, quadratic reciprocity, and the
classification of finite fields. Over 900 exercises are found throughout
the book.
