Selected Exercises in Algebra: Volume 1 (UNITEXT (119))

Description

This book, the first of two volumes, contains over 250 selected exercises in Algebra which have featured as exam questions for the Arithmetic course taught by the authors at the University of Pisa. Each exercise is presented together with one or more solutions, carefully written with consistent language and notation. A distinguishing feature of this book is the fact that each exercise is unique and requires some creative thinking in order to be solved. The themes covered in this volume are: mathematical induction, combinatorics, modular arithmetic, Abelian groups, commutative rings, polynomials, field extensions, finite fields. The book includes a detailed section recalling relevant theory which can be used as a reference for study and revision. A list of preliminary exercises introduces the main techniques to be applied in solving the proposed exam questions. This volume is aimed at first year students in Mathematics and Computer Science. Preface Preface to the English Edition Contents 1 Theory 1 Fundamentals 1.1 Sets 1.2 Maps 1.3 Relations 1.4 The Principle of Induction 1.5 Operations 1.6 Numbers 2 Combinatorics 3 Integers 3.1 Divisibility of Integers 3.2 Congruences 3.3 Modular Arithmetic 4 Groups 4.1 Definition and Basic Properties 4.2 Subgroups 4.3 Product of Subgroups 4.4 Cosets of a Subgroup 4.5 Normal Subgroups 4.6 The Symmetric Group 4.7 Group Homomorphisms 4.8 Direct Product of Groups 5 Rings 5.1 Definition and Basic Properties 5.2 Subrings, Ideals and Quotients 5.3 Polynomial Rings 5.4 Divisibility of Polynomials 5.5 Polynomial Factorisation 5.6 Quotients of Polynomial Rings 6 Fields 6.1 Characteristic of a Field 6.2 Multiplicative Group 6.3 Field Extensions 6.4 Splitting Field 6.5 Finite Fields 7 Preliminary Exercises 2 Exercises 1 Sequences 2 Combinatorics 3 Congruences 4 Groups 5 Rings and Fields 3 Solutions 1 Sequences 2 Combinatorics 3 Congruences 4 Groups 5 Rings and Fields Index