This book, the second of two volumes, contains approximately 350 exercises in Algebra which have featured exam questions for the Algebraic Structure and Algebra I courses 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 to be solved. The themes covered in this volume are: group theory and Sylow theorems, commutative rings with an emphasis on unique factorisation, Gaussian integers, field extensions and Galois theory. The book includes a detailed section recalling relevant theory that 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 second year students in Mathematics and Computer science.
Author(s): Rocco Chirivì, Ilaria Del Corso, Roberto Dvornicich
Series: UNITEXT, 140
Edition: 1
Publisher: Springer
Year: 2022
Language: English
Pages: 302
City: Cham
Tags: Algebra; Groups; Rings; Fields; Galois
Preface
Contents
1 Theory
1.1 Groups
1.1.1 Basic Notions
1.1.2 Homomorphism Theorems
1.1.3 Free Groups
1.1.4 Presentations of Groups
1.1.5 The Dihedral Group
1.1.6 Group Automorphisms
1.1.7 Commutators and Abelianisation
1.1.8 Group Actions
1.1.9 The Action by Conjugation
1.1.10 The Action by Multiplication
1.1.11 p-Groups
1.1.12 Permutations
1.1.13 Abelian Groups
1.1.14 Semidirect Product
1.1.15 Sylow Theorems
1.1.16 Simple Groups
1.2 Rings
1.2.1 Basic Notions
1.2.2 Maximal and Prime Ideals
1.2.3 Quotient Rings
1.2.4 Ideal Operations
1.2.5 Fraction Fields and Localisations
1.2.6 Divisibility
1.2.7 Euclidean Domains
1.2.8 Principal Ideal Domains
1.2.9 Unique Factorisation Domains
1.2.10 Gaussian Integers
1.3 Fields and Galois Theory
1.3.1 Basic Notions
1.3.2 Homomorphism Extensions
1.3.3 Galois Correspondence
1.3.4 Cyclotomic Extensions
1.3.5 Straightedge and Compass Constructions
1.4 Preliminary Exercises
2 Exercises
2.1 Groups
2.2 Rings
2.3 Fields and Galois Theory
3 Solutions
3.1 Groups
3.2 Rings
3.3 Fields and Galois theory
Index
