One option is Saunders Mac Lane and Garrett Birkhoff’s Algebra (3rd edition, AMS Chelsea, 1999). So suppose you want more than you’ll get from Beardon or perhaps you already have some - possibly fragmentary, possibly half-remembered - knowledge of algebra, and want to go up a level in sophistication and detail. But perhaps this doesn’t take you far enough to get a more rounded sense of modern algebra. At a really introductory level (first year undergraduate, perhaps), one really nice option which I can warmly recommend is Alan Beardon’s relatively short Algebra and Geometry (CUP, 2005), which is very well put together and indeed not-too-abstract. It depends what base you are starting from, of course. Fine: but what to read if - quite a big ‘if”! - you do decide to get to know more about this? One area which you have seen to be absolutely central to the modern mathematical curriculum is abstract algebra. As a philosopher of mathematics, you want to get to know more mathematics (after all, it is always a jolly good idea for a philosopher of X to know more than a mere smidgin about X). The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields.Aluffi, Algebra: Chapter 0 Suppose, e.g. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. An extensive set of exercises is presented at the end of each chapter. The algorithms developed for each topic are presented in a Pascal-like computer language. Numerous examples are integrated into the text as an aid to understanding the mathematical development. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions. The book first develops the foundational material from modern algebra that is required for subsequent topics. Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics.
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