This text covers most of the basic material on abstract algebra, with a more or less classical approach. The exposition is pitched about at the level of Birkhoff and MacLane, though the mathematical content is more extensive and slightly more sophisticated. The book starts with set theory, a smattering of number theory, and then plunges into a short course on finite groups, ending with the existence of Sylow groups. A few commutative diagrams are visible. Next come rings and ideals, fields of quotients, polynomial rings, vector spaces and modules, including dual spaces, inner product spaces and the fundamental theorem on finitely generated modules over Euclidean rings (without uniqueness statements). Then the elementary theory of fields (surprise: a proof of the transcendence of e), ruler and compass constructions, Galois theory and solvability by radicals. Almost the last third of the book is devoted to linear transformations and quadratic forms. The Jordan canonical form is done by reduction to nilpotent transformations and direct decomposition of the vector space; this is followed by the rational canonical form, done independently as a consequence of the fundamental theorem on modules over a polynomial ring. Then come traces, transposes (of matrices, dual spaces do not appear here), determinants; Hermitian, unitary and normal transformations; real quadratic forms. Finally, there is a short chapter on three topics that have led to extensive mathematical research: Wedderburn's theorem on finite division rings, the classification of all (i.e., all three) real division algebras, and the four squares theorem (beginning of Waring's problem). One gets the impression that in writing this text, the author was not interested in the content of the theorems alone, but also in the mathematical experience to which the student would be exposed, thus explaining several deliberate instances of inefficient proofs (the most significant was mentioned above: the independent reductions to Jordan and rational canonical form, without deriving either as a corollary of the other).
Reviewer: Zelinsky, D. [form MathSciNet]