Textbook:
- " A first course in abstract algebra" by
J. J. Rotman, Prentice Hall, 2000.
Supplementary Books:
- "Contemporary Abstract Algebra" by J. A. Gallian,
D. C. Heath and Company, 1994.
- "Algebra: Abstract and Concrete" by F. M. Goodman,
Prentice Hall, 1998.
- "Fundamentals of Abstract Algebra" by D. S. Malik, J. N.
Mordeson, M. K. Sen, McGraw-Hill, 1997.
- "Abstract Algebra" by T. Shifrin, Prentice Hall, 1996.
- "Algebra" by M. Artin, Prentice Hall, 1991.
If these books are difficult for you, then before attacking them see:
- "A First Course in Abstract Algebra" by J. B. Fraleigh,
Addison-Wesley.
Graduate (hence more difficult!) Algebra Texts:
- "Abstract Algebra" by Dummit and Foote, Prentice Hall, 1991.
- "Algebra" by T. W. Hungerford, Springer, 1974.
- "Algebra, a Graduate Course" by I. M. Isaacs, Brooks/Cole Publishing
Company, 1994.
- "Algebra" by S. Lang, Addison-Wesley, 1993.
More About Groups:
- "Fundamentals of the Theory of Groups" by M. I. Kargapolov
and J. I. Merzljakov, Springer.
- "A Course in the Theory of Groups" by D. Robinson, Springer, 1996.
More About How Algebra and Geometry can be Combined:
- "Glimpses of Algebra and Geometry" by G. Toth, Springer, 1998.
- "Finite Reflection Groups" by Benson and Grove, Springer.
- "Groups and Symmetry" by M. A. Armstrong, Springer, 1988.
- "Adventures in Group Theory" by D. Joyner, Johns Hopkins,
2002.
More About Rings:
- "Rings, Modules and Linear Algebra" by B. Hartley and T. O. Hawkes,
Chapman and Hall, 1970.
- "Introductory Lectures on Rings and Modules" by J. A. Beachy,
Cambridge University Press, 1999.
More About Fields:
- "Field Theory" by S. Roman, Springer.
If you are looking for challenging problems at the undergraduate level,
then see the exercises in:
- "Topics in Algebra" by I. N. Herstein, Wiley and Sons.
What about some history?
- "A History of Mathematics" by Carl B. Boyer.
- "Mathematical Thought from Ancient to Modern Times" by M. Kline.
- "A Concise History of Mathematics" by D. J. Struik.