Selected Exercises in Algebra Volume 1

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.

1 Theory review.- 1.1 Fundamentals.- 1.2 Combinatorics.- 1.3 Integers.- 1.4 Groups.- 1.5 Rings.- 1.6 Fields.- 2 Exercises.- 2.1 Sequences.- 2.2 Combinatorics.- 2.3 Modular arithmetic.- 2.4 Groups.- 2.5 Rings and Fields.- 3 Solutions.- 3.1Sequences.- 3.2 Combinatorics.- 3.3 Modular arithmetic.- 3.4 Groups.- 3.5 Rings and Fields.

Rocco Chirivì obtained a degree in Mathematics from the University of Pisa in 1995; he earned his Diploma di Licenza from the Scuola Normale of Pisa in 1997, followed by his Diploma di Perfezionamento in 2000. He was a researcher in Algebra at the University of Pisa from 2002 to 2012 and has since been working as a researcher at the University of Salento. His main research focus is Representation Theory, at the intersection between Algebra, Combinatorics and the geometry of varieties related to group actions.

Ilaria Del Corso attended the Corso di Perfezionamento at the Scuola Normale from 1990 to 1992 and has been an associate professor in Algebra at the University of Pisa since 2001. She has extensive teaching experience in Algebra and Algebraic Number Theory. Her research in Algebraic Number Theory concerns number fields and local fields, focusing in particular on the study of ramification and on the Galois module structure of certain field extensions.

Roberto Dvornicich, having been a student at the Scuola Normale of Pisa, obtained a degree in Mathematics from the University of Pisa in 1972 and later attended the Corso di Perfezionamento at the Scuola Normale. He has been a full professor in Algebra at the University of Pisa since 1990. His main research interests lie within Algebraic Number Theory (arithmetic properties of number fields and local fields) and in Diophantine Analysis (algebraic equations over the integers or some number field).