Titlebook: Lectures in Abstract Algebra I; Basic Concepts Nathan Jacobson Textbook 1951 Nathan Jacobson 1951 Galois theory.Permutation.algebra.automor

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Introduction: Concepts from Set Theory the System of Natural Numbers,ces. The study of these systems encompasses a major portion of classical algebra. Thus, in a sense our subject matter is old. However, the axiomatic development which we have adopted here is comparatively new. A beginner may find our account at times uncomfortably abstract since we do not tie oursel

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Rings, Integral Domains and Fields,cted binary compositions. Unlike the theory of groups which had essentially one source, namely, the study of sets of 1-1 transformations relative to resultant composition, the theory of rings has been fused out of a number of special theories. For this reason it will appear to be somewhat less unifi

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Extensions of Rings and Fields,r ring that has the required properties. Thus, for example, there exist equations of the form . = . ≠0 that have no solutions in the domain of integers. The field of rational numbers is constructed for the purpose of insuring the solvability of equations of this type. The method used to construct th

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Elementary Factorization Theory,r of important integral domains such factorizations exist for all the non-units, and in a certain sense uniqueness of factorization holds. In these instances we can determine all of the factors of a given element and hence we can give simple conditions for the solvability of equations of the form .

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Groups with Operators,d those of a homomorphic image, normal series and composition series, the Schreier theorem, direct products and the Krull-Schmidt theorem. The range of application of these results is enormously extended by introducing the new concept of a group with operators. This concept, which was first consider

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