Jacobson N. Basic Algebra II

Dover Publications, 2009. - 704 Pages.Volume II of a pair of classic texts — and standard references for a generation — this book comprises all of the subjects of first-year graduate algebra. In addition to the immediate introduction and constant use of categories and functors, it revisits many topics from Volume I with greater depth.The most extensive changes in this edition occur in the segment of the book devoted to commutative algebra, especially in Chapter 7, Commutative Ideal Theory: General Theory and Noetherian Rings; Chapter 8, Field Theory; and Chapter 9, Valuation Theory. In Chapter 7 we give an improved account of integral dependence, highlighting relations between a ring and its integral extensions ("lying over, " "going-up, " and "going-down" theorems). Section 7.7, Integrally Closed Domains, is new, as are three sections in Chapter 8: 8.13, Transcendency Bases for Domains; 8.18, Tensor Products of Fields; and 8.19, Free Composites of Fields. The latter two are taken from Volume III of our Lectures in Abstract Algebra (D. Van Nostrand 1964; Springer-Verlag, 1980). The most notable addition to Chapter 9 is Krasner's lemma, used to give an improved proof of a classical theorem of Kurschak's lemma (1913). We also give an improved proof of the theorem on extensions of absolute values to a finite dimensional extension of a field (Theorem 9.13) based on the concept of composite of a field considered in the new section 8.18.