Book review: Charles C. Sims, Abstract Algebra - A Computational Approach

This text is writ ten for a one year undergraduate introduction to abs t rac t algebra. Its novelty lies in a consistent use throughout of APL for examples and algebraic computations. An extensive background in programming is not a prerequisite for either instruction or for students, but the author suggests a minimum of three weeks for classroom discussion at the outset to familiarize the student with APL. A separate APL text is not necessary since a ,50 page introduction is included as an appendix. However two workspaees, CLllSSLIB and EXA~,fPLES, available in machine readable form from the author, are necessary for the use of the text. CL.aSSLIB contains over 200 APL functions ranging from a function to compute all orbits of a permutat ion group to functions for computing irreducible and prime elements in special Euclidean domains. A separate instructor 's manual and a CLASSLIB user's manual is available from the publisher. After preliminary chapters on sets and the integers, a chapter on group theory provides quite s tandard material and concludes with the Sylow theorems. More tradit ional exercises are provided after each section of text along with exercises involving APL functions. For the most part , traditional notat ion is used in proofs and exposition while the APL character set appears in examples and exercises. The chapter on ring theory ends with a discussion of PIO's and UFD's with APL being used in computations with arrays whose entries are in such rings as Z, Z,~, Q, .dT:g, e , and ~[ i ] . Chapters on modules, modules over Euclidean domains, and linear transformations provides an introduction to theoretical linear algebra. There is a brief chapter on field extensions. Calculations in finite fields and factorization in Zp[x I are t reated using APL but the text contains no Galois theory, unfortunately. Sims' book is a serious text which has been developed over a number of years and deserves to be widely used. L.J. Cummings Dept. of Pure Mathematics University of Water loo