Abstract algebra / Chih-han Sah

The text begins with the seemingly obligatory Chapter 0 outlining the usual set-theoretic preliminaries and, in particular, giving a brief treatment of Zorn's lemma. Chapter I develops the natural numbers, rational integers and rational numbers from the foundation of the Peano axioms. The elementary properties of divisibility and unique factorization in the rational integers are given. In Chapter II groups, rings, integral domains and fields are defined as are their morphisms. An unusual touch here is a discussion of the Frobenius homomorphisms for commutative rings of finite characteristic. In Chapter III some of the standard theorems about groups are proved, including Wielandt's attractive proof of the main Sylow theorem. A discussion of the finite symmetric groups is given and simplicity of the larger alternating groups is proved. The chapter ends with a proof of the structure theorem for finite abelian groups. This is done directly and independently of the structure theorem for finitely generated modules over a principal ideal domain. Chapter IV discusses some of the elementary properties of rings, including principal ideal domains and unique factorization domains. The chapter ends with a consideration of various properties of the endomorphism ring of a module. In Chapter V the theory of modules is entered into and the structure theorem for finitely generated modules over a principal ideal domain is given. The chapter also contains a treatment of tensor products, tensor algebras, exterior algebras, and duality theory. Chapter VI is a reasonably traditional treatment of vector spaces including the usual canonical form theorems. In Chapters VII and VIII the theory of fields and Galois theory are covered. (MathSciNet)

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