What are the topics covered in I.N. Herstein's Topics in Algebra?

[micromass]

• Author: I.N. Herstein
• Title: Topics in Algebra
• Amazon Link: https://www.amazon.com/dp/0471010901/?tag=pfamazon01-20
• Prerequisities:
• Level: Grad

Table of Contents:

[LIST]
[*] Preliminary Notions
[LIST]
[*] Set Theory
[*] Mappings
[*] The Integers
[/LIST]
[*] Group Theory
[LIST]
[*] Definition of a Group
[*] Some Examples of Groups
[*] Some Preliminary Lemmas
[*] Subgroups
[*] A Counting Principle
[*] Normal Subgroups and Quotient Groups
[*] Homomorphisms
[*] Automorphisms
[*] Cayley's Theorem
[*] Permutation Groups
[*] Another Counting Principle
[*] Sylow's Theorem
[*] Direct Products
[*] Finite Abelian Groups
[/LIST]
[*] Ring Theory
[LIST]
[*] Definition and Examples of Rings
[*] Some Special Classes of Rings
[*] Homomorphisms
[*] Ideals and Quotient Rings
[*] More Ideals and Quotient Rings
[*] The Field of Quotients of an Integral Domain
[*] Euclidean Rings
[*] A Particular Euclidean Ring
[*] Polynomial Rings
[*] Polynomials over the Rational Field
[*] Polynomial Rings over Commutative Rings
[/LIST]
[*] Vector Spaces and Modules
[LIST]
[*] Elementary Basic Concepts
[*] Linear Independence and Bases
[*] Dual Spaces
[*] Inner Product Spaces
[*] Modules
[/LIST]
[*] Fields
[LIST]
[*] Extension Fields
[*] The Transcendence of e
[*] Roots of Polynomials
[*] Construction with Straightedge and Compass
[*] More About Roots
[*] The Elements of Galois Theory
[*] Solvability by Radicals
[*] Galois Groups over the Rationals
[/LIST]
[*] Linear Transformations
[LIST]
[*] The Algebra of Linear Transformations
[*] Characteristic Roots
[*] Matrices
[*] Canonical Forms: Triangular Form
[*] Canonical Forms: Nilpotent Transformations
[*] Canonical Forms: A Decomposition of V: Jordan Form
[*] Canonical Forms: Rational Canonical Form 
[*] Trace and Transpose
[*] Determinants
[*] Hermitian, Unitary, and Normal Transformations
[*] Real Quadratic Forms
[/LIST]
[*] Selected Topics
[LIST]
[*] Finite Fields
[*] Wedderburn's Theorem on Finite Division Rings
[*] A Theorem of Frobenius
[*] Integral Quaternions and the Four-Square Theorem 
[/LIST]
[/LIST]

 

[mathwonk]

Years ago I read a lot of this book and worked a lot of the problems. I think I benefited more from the problems, although now that I understand more I am not sure why. The explanations and proofs are very strange in the following way: they go slowly and clearly and make every step seem absolutely irresistible. then at the end three very odd things are true that seem almost incompatible:
1) you are completely convinced the argument was correct,
2) you cannot remember any of it,
3) you have no intuitive insight at all into why the result is true.

I suspect this is because he was writing for a sophomore honors class about material usually taught later. so he chose his results carefully as ones that could be covered step by step without too much background, and he chose his method of presentation to be as easy to present as possible, without actually shedding any light on the underlying phenomena.

Unfortunately this makes the book almost useless to a future mathematician, since after reading it and beginning grad school, I did not really understand anything and had to learn it all over again elsewhere. Another annoying thing is he uses the "algebraists" convention on notation, which is opposite to everyone else in the world, matrices are multiplied on the wrong side, etc etc...

Now however I admit I do look in there very occasionally to see the structure of an orthogonal matrix or some such, or maybe a proof of the theorem on representing integers as sums of 4 squares,... but i never try to learn anything out of it.

 

[mpresic]

I read most of this book out of interest although it was never required. The only trouble with the book is in the preface he says many of the problems may not be doable with the material he presented at the time he poses the problem. If you cannot get the problem (I could not get most problems), you are never sure that you could be expected to get the problem or not.

I know the book is highly recommended by mathematicians. I also like his warnings like (paraphrased) understand this now because in a few sections it will be too late.

He really gets into Sylow Theorems (as he says in the preface). Enjoyed his discussion of finite groups

 

[mathwonk]

the mathematician who posted just before you did not recommend it highly.

this mathematician recommends mike artin's book, Algebra.

 

[mpresic]

He may be right. I do have a copy of Artin's book as well. Topics in Algebra is a much older book.
I almost regard Herstein's problems as a curse. As I said, I was never sure some of his difficult problems could be done with the text information that was presented so far. I think other of his problems were doable but with material from other texts (like Artin) presented in a different order.

The reason I may feel this way is because this was my real mathematics course where proof-writing was essential. I am a physicist not a mathematician and I think algebraicists have a certain facility with concepts that I find difficult. Some mathematician friends of mine say they have similar feelings towards many physics problems.

 

[gianeshwar]

Dear Physicist friends! Please see my post in Algebra Forum.