Algebra – Thomas W. Hungerford

This text is a classic in graduate-level algebra. Algebraists tend to be quite fond of this book, though I would not give it a favorable review. First, I’ll note its positive points: it makes a great reference. This text is an extremely dense (forgive the math pun) treatment of abstract algebra, with very compact theorems. It’s extremely comprehensive: one could easily spend a couple semesters studying the material in this book. Honestly, that’s about all the positive I can say. While proofs done in as few lines as possible are admirable, they’re hardly illuminating to a student trying to learn difficult concepts. Some of the proofs aren’t really even proofs; rather, they’re sketches with a great many details missing. Moreover, the text lacks illustrative examples to really help a reader understand the concepts, rather than regurgitate definitions and apply them. I also dislike definitions that require references to other definitions, and theorems that reference other theorems, i.e. (This is true by Theorem 3.5(iii)).  Obviously nothing in this text is wrong, but it’s not particularly helpful to a newer student. The exercises are decent. I would not recommend this for any level of self-study, nor would I recommend this to teach from (though I realize it’s likely one of the most common used in universities.) If one is already an algebraist, the book may be considered readable, but the goal of a mathematical text is to educate those who are are not necessarily experts in its specific subfield (again, pardon the math pun). In summary, Hungerford created a great reference, but I don’t recommend it for much else. 

-Rachel Traylor, Ph.D.

 

I second all points made by Dr. Traylor above, having used this book as a text for a first course in graduate algebra.  As a reference, this book works well, with remarkable subject coverage.  Further, Hungerford does keep to a plan outlined at the outset, one which may help motivate some students.  The standout weakness of the book (for me personally) is the reliance on numerical reference to previous–and future–parts of the book.  Many proofs of theorems and corollaries are ‘sketched’, e.g. merely stating “Corollaries I.4.6 and 5.3 and Theorem  5.7” or pointing the reader to an exercise.  This is less than helpful.  For the experienced reader, I recommend this solely as a quick reference.  For a newer student to the subject or general non-algebraist, I suggest Pinter’s A Book of Abstract Algebra.

 

-Jason Hathcock

Review

Prerequisites

very strong undergraduate abstract algebra


Topics Covered

  • Basics: sets, equivalence classes, functions, relations
  • Group Theory
    • semigroups, monoids, groups
    • homomorphisms and subgroups
    • cyclic groups
    • cosets
    • normality and quotient groups
    • symmetric and dihedral groups
    • products, coproducts, and free objects
    • direct products/sums
    • free groups, free products, generators, and relations
  • Group Structures
    • Free Abelian groups
    • Finitely generated abelian groups
    • action of a group
    • Sylow Theorems
  • Rings
    • Ideals
    • Factorization
    • Rings of Quotients
    • Formal Power Series
  • Modules
    • free modules, vector spaces
    • Hom and Duality
    • Tensor Products
  • Fields and Galois Theory
  • Structure of Fields
  • Linear Algebra
    • Matrices/maps
    • Rank/Equivalence
    • Determinants
  • Commutative Rings/Modules
    • Primary Decompositions
    • Dedekind Domains
  • Ring Structure
  • Categories
    • Functors and Natural Transformations
    • Adjoint Functors
    • Morphisms

Attributes

Difficulty 5
Good for teaching? 2
Proof Quality 1
Quality of Exercises 3
Readability 1
Self-Study 1