Stage 4 Algebra

Algebra:
Materials mentioned in stage 3, which have not been covered there, can be studied at this stage, for example, Artin, Dummit and Foote.

My lecturer Daniel Chan's web page contains lots of algebra stuffs.

In general:

• Artin E. Notes by A.A. Blank Modern Higher Algebra (Galois Theory)
• Ash R.B. Abstract Algebra: The Basic Graduate Year (FREE!)
• Bourbaki N. Elements of Mathematics: Algebra I, II - Nicolas Bourbaki is a group of French mathematician aimed to write rigour mathematic which based on set theory. Not recommended for a beginner, but if you like, search around for topics.
• Burris S.N. and Sankappanavar H.P. A Course in Universal Algebra The Millennium Edition (FREE!)
• Cohn P.M. Algebra, Vol 1, 2, 3
• Garrett P. Abstract Algebra (FREE!) - Over 400 pages.
• Grillet P.A. Abstract Algebra
• Herstein I.N. Topics in Algebra
• Hungerford T.W. Algebra - James Wilson has a solutions manual [.pdf].
• Isaccs I.M. Algebra: A Graduate Course - Non-standard treatment.
• Jacobson N. Lectures in Abstract Algebra I, II, III - Classic.
  
• Jacobson N. Basic Algebra II - Originally envisioned as new editions of his Lectures, the main difference is that category theory was not yet developed when he was preparing the previous version.
• Lang S. Algebra - My first Springer GTM. Only a few of us dare to ignore it. Ignoring it may look cool, but I'm not going to do so. To quote an Amazon customer, '[t]his will teach you how to run if you know how to walk.' Some readers think that if one wanna study algebraic number theory, one should be comfortable with Lang's Algebra first. See George M. Bergman's Companion.
• Sather-Wagstaff S. Graduate Algebra Notes [.pdf] (FREE!)
• Smith R. Algebra Course Notes (FREE!)
• Spindler K. Abstract Algebra with Applications Vol I, II
• Surowski D. Workbook in Higher Algebra [.pdf] (FREE!)
• Rotman J.J. Advanced Modern Algebra - See author's web for errata.
• van der Waerden B.L. Algebra: Volume I, II - For historical interest, perhaps.
• Wickless W.J. A First Graduate Course in Abstract Algebra
• Wilkins D.R. Abstract Algebra (FREE!)

Advanced Linear Algebra:

• Golan J.S. The Linear Algebra - A Beginning Graduate Student Ought to Know
• Greub W.H. Linear Algebra
• Horn R.A. and Johnson C.R. Matrix Analysis
• Matthews K.R. and Fama C. Linear Algebra Notes (FREE!)
• Roman S. Advanced Linear Algebra

Groups and Lie Algebras:

• Alperin J.L. and Bell R.B. Groups and Representations
• Baker A. Matrix Groups: An Introduction to Lie Group Theory
• Chevalley C. Theory of Lie Groups
• Erdmann K. and Wildon M.J. Introduction to Lie Algebras
• Hall B.C. An Elementary Introduction to Groups and Representations (FREE!)
• Hall B.C. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction - See his web for information.
• Humphreys J.F. A Course in Group Theory
• Humphreys J.F. Linear Algebraic Groups
• Jacobson N. Lie Algebras
• Kurzweil H. and Stellmacher B. The Theory of Finite Groups: An Introduction
• Milne J.S. Algebraic Groups and Arithmetic Groups (FREE!)
• Lang S. SL₂(R)
• Rossmann W. Lie Groups: An Introduction through Linear Groups
• Rotman J.J. An Introduction to the Theory of Groups
• Samelson H. Notes on Lie Algebras [.pdf] (FREE!)
• Schmidt O.U. Abstract Group Theory
• Sepanski M.R. Compact Lie Groups
• Serre J.-P. Linear Representations of Finite Groups
• Serre J.-P. Trees - More advanced work.
• Skorobogatov A.N. Lie Algebras Lecture Notes [.pdf] (FREE!)
• Sternberg S. Lie Algebras [.pdf] (FREE!)
• Tauvel P. and Yu R.W.T. Lie Algebras and Algebraic Groups
• Varadarajan V.S. Lie Groups, Lie Algebras, and Their Representation
• Weyl H. The Classical Groups: Their Invariants and Representations

Rings, Fields and Galois Theory:
Note several stage 3 algebra texts also introduce Galois Theory. For example, J.B. Fraleigh's one.

• Artin E. Galois Theory (here for FREE!)
• Baker A. Galois Theory [.pdf] (FREE!)
• Borceux F. and Janelidze G. Galois Theories
• Edwards H.M. Galois Theory
• Escofier J.-P. Galois Theory - Assumed knowledge is standard algebra, i.e. stage 3 algebra above. Despite the nice treatment of the subject itself, the author also gives historical remarks for the interesting life of Galois.
• Ghorpade S.R. Lectures on Field Theory and Ramification Theory [.pdf] (FREE!)
• Ghorpade S.R. Notes on Galois Theory [.pdf] (FREE!)
• Hadlock C.R. Field Theory and Its Classical Problems
• Herstein I.N. Noncommutative Rings
• Howie J.M. Fields and Galois Theory
• Lam T.Y. A First Course in Noncommutative Rings
• Lam T.Y. Exercises in Classical Ring Theory
• Milne J.S. Fields and Galois Theory (FREE!)
• Morandi P. Field and Galois Theory
• Roman S. Field Theory
• Stewart I. Galois Theory - Another well written book on this subject.
• Swallow J. Exploratory Galois Theory
• Weintraub S.H. Galois Theory - See author's website for errata.
• 李華介：簡介 Galois 理論 (FREE!)

Modules and Representation Theory:

• Fulton W. and Harris J. Representation Theory: A First Course
• Ghorpade S.R. and Verma J. K. Primary Decompositions of Modules [.pdf] (FREE!)
• Humphreys J.E. Introduction to Lie Algebras and Representation Theory
• Kirillov A.A. Elements of the Theory of Representations
• Lam T.Y. Lectures on Modules and Rings
• Li W.C.W Lectures on Modular Forms and Galois Representations [.pdf] (FREE!)
• Woit P. Lie Groups and Representations (FREE!)

Commutative Algebra:

• Ash R.B. A Course In Commutative Algebra (FREE!)
• Atiyah M.F. and MacDonald I.G. Introduction to Commutative Algebra (here is a scanned version for FREE!) - Solutions by Jinhyun.
• Cox D.A., Little J.B. and O'Shea D.B. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra - See their web for typo, etc.
• Eisenbud D. Commutative Algebra: with a View Toward Algebraic Geometry
• Fesenko I. Introduction to Commutative Algebra [.pdf] (FREE!)
• Ghorpade S.R. A Quick Review of Commutative Algebra [.pdf] (FREE!)
• Ghorpade S.R. Lectures on Commutative Algebra [.pdf] (FREE!)
• Reid M. Undergraduate Commutative Algebra
• Zariski O. and Samuel P. Commutative Algebra

Homological Algebra and Category:

• Adámek J., Herrlic H. and Strecker G.E. Abstract and Concrete Categories: The Joy of Cats (FREE!)
• Awodey S. Category Theory
• Barr M. and Wells C. Toposes, Triples and Theories (here for FREE!)
• Cartan H. and Eilenberg S. Homological Algebra
• Fesenko I. Introduction to Homological Algebra [.pdf] (FREE!)
• Fokkinga M.M. A Gentle Introduction to Category Theory: The Calculational Approach (FREE!)
• Freyd P. Abelian Categories (FREE!) - This site consists dozens of Reprints in Theory and Applications of Categories for further reading.
• Hilton P.J. and Stammbach U. A Course in Homological Algebra
• Kashiwara M. and Schapira P. Categories and Sheaves
• Lawvere F.W. and Schanuel S.H. Conceptual Mathematics: A First Introduction to Categories
• Mac Lane S. Categories for the Working Mathematician
• Mac Lane S. Homology
• Rotman J.J. Introduction to Homological Algebra
• Sather-Wagstaff S. Homological Algebra Notes [.pdf] (FREE!)
• Weibel C.A. An Introduction to Homological Algebra - Correction here. Solution by Jinhyun.