Kaiming Zhao – författare

This book is intended as a textbook for a one-term undergraduate second linear algebra course which prepares students for both applied mathematics and pure mathematics, in particular, for Galois Theory. The students are assumed to have some knowledge on calculus, linear systems, determinates and matrices. Earlier drafts of this book have been used as the textbook when the author teaches MA222, the second linear algebra course of the three, at Wilfrid Laurier University since 2019.The book contains 196 exercises of varying difficulty which will allow students to practice their own computational and proof-writing skills. Detailed solutions, answers or hints to all the exercises are provided in the book. Besides standard ones, many of the exercises are provided in the book. Besides standard ones, many of the exercises are interesting. Some are rather hard. It is not a surprise if the reader cannot solve some of the exercises by themselves, particularly for first learners. It is strongly suggested that the readers first try to solve them without looking at the solutions of hints.

Group Theory is intended as a textbook for a one-term course in group theory for senior undergraduate or graduate students. It provides students with good insight into group theory as quickly as possible. Simplicity and working knowledge are emphasized here over mathematical completeness. This book will provide a rigorous proof-based modern treatment of the main results of group theory. As a result, proofs are in great details with a lot of interesting examples.This book can also serve as a reference for professional mathematicians. The book contains 221 carefully selected exercises of varying difficulty which will allow students to practice their own computational and proof-writing skills. Sample solutions to some exercise questions are provided, from which students can learn to write their own solutions and proofs. Besides standard ones, many of the exercises are very interesting.Group Theory includes:basic properties of groupssymmetric groups and alternating groupsisomorphism theorems, commutator subgroups, direct and semi-direct products of groupsgroup action on sets, Cauchy-Frobenius Lemma, symmetries of the regular polyhedronsJordan-Holder Theorem and Sylow Theoremsfree groups, group presentations, finitely generated abelian groupssolvable groups, Hall subgroups, characteristically simple groups, Sylow systems and Sylow basesnilpotent groups, Frattini subgroupsbasic properties of representations of groups on vector spaces, subrepresentations; irreducibility, Schur's Lemma, and Maschke's Theorem

This book is intended as a textbook for a one-term senior undergraduate (or graduate) course in Ring and Field Theory, or Galois theory. The book is ready for an instructor to pick up to teach without making any preparations.The book is written in a way that is easy to understand, simple and concise with simple historic remarks to show the beauty of algebraic results and algebraic methods. The book contains 240 carefully selected exercise questions of varying difficulty which will allow students to practice their own computational and proof-writing skills. Sample solutions to some exercise questions are provided, from which students can learn to approach and write their own solutions and proofs. Besides standard ones, some of the exercises are new and very interesting. The book contains several simple-to-use irreducibility criteria for rational polynomials which are not in any such textbook.This book can also serve as a reference for professional mathematicians. In particular, it will be a nice book for PhD students to prepare their qualification exams.