This is good basic introduction to abstract algebra at the undergraduate level. It is more concrete than most such books, and is unusual in starting with rings first, making heavy use of the integers and polynomial rings as guides, and then doubling back to groups. This third edition has added a little bit of alternate treatment of some topics to provide a path starting with group theory.
The book has a strong number-theoretic flavor, with the exception of the field theory parts, which have a strong theory-of-equations flavor. I think this rings-first approach works well; although groups are simpler, the integers are much more familiar and it’s easier to go through the abstraction process.
This is not primarily a proofs book, and it recognizes the possibility that this may be the first course where students have to prove things. The book shows many examples to motivate the theorem and make it plausible, and then gives a proof. The book is well-equipped with exercises, although most of these are drill, examples, and proofs that test understanding rather than pose challenging problems or provide further developments of ideas in the body.
Very Good Feature: extensive symbol glossary and index printed on the endpapers. Very Peculiar Feature: three Technology Tips scattered through the book showing how to use a TI graphing calculator for some number theory calculations. The programs are available for download from the publisher’s web site.
Two comparable books are Herstein’s Abstract Algebra and Fraleigh’s A First Course in Abstract Algebra. Both of these follow the more traditional groups-rings-fields ordering. Herstein’s book is even more basic than Hungerford’s and is half the length. In particular, it omits Galois theory and solvable groups, although it does show the beginning of Galois theory and develops enough about field extensions to handle the classic construction problems in geometry. Fraleigh is fairly close to Hungerford in coverage, but is generally more abstract, puts more emphasis on structure and homomorphisms, has more applications, and goes a little deeper into the subject. Hungerford does a lot more hand-holding that the other two books, and sets a very leisurely pace, which accounts for much of the length of his book.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.
1. Arithmetic in Z Revisited.
2. Congruence in Z and Modular Arithmetic.
4. Arithmetic in F[x].
5. Congruence in F[x] and Congruence-Class Arithmetic.
6. Ideals and Quotient Rings.
8. Normal Subgroups and Quotient Groups
9. Topics in Group Theory.
10. Arithmetic in Integral Domains.
11. Field Extensions.
12. Galois Theory.
13. Public-Key Cryptography.
14. The Chinese Remainder Theorem.
15. Geometric Constructions.
16. Algebraic Coding Theory.
17. Lattices and Boolean Algebras (available online only).