Selected Exercises in Algebra

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).

“This book is filled with rich exercises with detailed historical notes mirroring Arithmetic and Algebra. … This book can be used for both the professor who needs a reservoir of problems for homework or projects and for the student who wishes to tackle more exercises for drill and practice. The book left me thinking that the more you practice and know, the better.” (MAA Reviews, April 12, 2020)

• 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.