Additional References
Table of Contents
On this page you will find additional references recommended by professors and senior students. They cover a wide range of topics and vary in nature (descriptive, expository, advanced, generic). We plan to add more resources as recommended by professors and research scholars.
Galois Theory and Beyond #
We are sure that everyone would agree that Galois theory is perhaps the most beautiful and exciting topic of Algebra covered in the four major compulsory Algebra courses at B. Math. Not only is the theory robust, interesting and applicable, it has a really wonderful and adventurous history to it. The robustness of the theory is so much so that it can be easily applied to other avenues of math like Topology (Topological Galois Theory), Analysis (Differential Galois Theory), Differential Equations (Galois Theory of Differential Equations), Algebraic Equations (Galois Theory of Algebraic Equations), Connections and Lattices (Galois Connections and Lattices)… the list goes on.
Here we present a few materials which can be referred to by readers interested to know more about the history or myriad arenas where this wonderful concept of attaching groups and analysing more complex objects with the same can be extended. Note that we assume basic Familiarity with Fields and Galois Theory (if you are not, do check out the wonderful texts by J M. Howie (Fields and Galois Theory), Dummit and Foote (Abstract Algebra, Chapter 13, 14) and this old but wonderfully written book by J. J. Rotman (Galois Theory)).
Fields and Galois Theory by Patrick Morandi: Sometimes used as a textbook in many courses, this wonderful book not only deals with the basic concepts of Galois Theory from the scratch but also develops the same much further to analyse Cyclic and Cyclotomic Extensions, Finite Fields, Group Cohomology, Algebraic Expressions, Independences and Varieties, Infinite Galois as well as Transcendental Expressions.
Galois Theory by Harold M. Edwards: This book beautifully deals with the original ideas presented in Galois’ Original Memoir. So many proofs are different, more basic and also the theory is intertwined with several anecdotes, historical references and even interesting Do-It-Yourself problems.
Abel’s Theorem in Problem and Solutions by V. B. Alekseev: Though some may find the first part of the book a standard compendium of problems and cheat-sheets in Groups and Complex Variables, the book is inspired and wonderfully presents scopes to apply Galois Theory, especially Abel’s Theorem, to Algebraic equations of Complex variables. Also, the Appendix by A. Khovanskii is one-of-its-kind, in the sense it actually introduces concepts from Topology (General, Algebraic and Differential) and merges it smoothly with the theory in hand.
Galois’ Dream, Group Theory and Differential Equations by Michio Kuga: Originally written in Japanese, this book is a unique blend of theory, history, anecdote and applications. Not only does it deal with Galois Theory, it introduces concepts from Algebraic Topology with a view towards Galois Theory and slowly approaches Function Theory of Covering Surfaces and Differential Equations - asking for solvability and also other interesting facts, by attaching groups (familiar, eh?)
Galois Theory by Jean-Pierre Escofier: This is a very lucid and somewhat standard textbook at first sight. However, a closer look tells us that many new methods, historical developements in chronology, tales and variants of the problem in hand as well as some fun facts are presented throughout the book, thus making the book a really interesting and enlightening read.
Galois’ Theory of Algebraic Equations by Jean-Pierre Tignol: Starting from the what some may call the pre-history of Algebra, this book slowly brings to context Polynomials, their origin, solutions, developments in the theory and solution of Algebraic Equations and cleverly “concludes” with Galois Theory. Curious, isn’t it? So why not give a read!
(Thanks to Professor Jishnu G. Biswas for suggesting such a wonderful collection of texts)
Some Notes on Algebra and more #
J. S. Milne has a wonderful collection of lecture notes for several different topics on his website. Any advanced undergraduate, postgraduate or even research student is highly recommended to have a glance at them. Same as Conrad, they give a flavour of research, note-making and proof writing. He has very detailed notes ranging from basic group theory to very abstract and actively researched topics such as Reductive Groups, Algebraic Groups, Complex Multiplication, etc.
Refer to the following link
Keith Conrad has a wonderful collection of expository notes on his websites. Any undergraduate (as well as Post Graduate) student is highly recommended to have a glance at them. Not only will it help in learning the topic at a deeper level, but will also give a flavour of research, note-making and proof writing.
Refer to the following link. He has also included notes on advanced algebra topics, Number Theory, Analysis and Topology.
(Thanks to Prof. Anita Naolekar for recommending this.)
Roots to Research – A Vertical Development of Mathematical Problems #
Math students may find the following book helpful.
- Title: Roots to Research – A vertical development of Mathematical Problems
- Authors: Judith D. Sally & Paul D. Sally, Jr.
- Topic: Introduction to Research
https://bookstore.ams.org/mbk-48/
(Thanks to Prof. D. Yogeshwaran for recommending this.)
Thirty-three Miniatures – Mathematical and Algorithmic Applications of Linear Algebra #
The following book may be useful for math students.
- Title: Thirty-three Miniatures – Mathematical and Algorithmic Applications of Linear Algebra
- Authors: Jiří Matoušek
- Topics: Computational Linear Algebra, etc.
https://bookstore.ams.org/stml-53
(Thanks to Prof. D. Yogeshwaran for recommending this.)