I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas:

a) Commutative Algebra

b) Field Theory and Galois Theory

c) Homological Algebra

My question is given that I have knowledge in the above topics which is the next field that is appropriate to pick up to continue my studies in Algebra. This I feel is a relavent question because its always fun to pick up a field that is just out your reach and try to understand and master it, this is where I need help.

When you suggest a field keep in mind that it should just be out of reach in terms of abstractness, and please specify weather the suggested field actually applies the above topics or is a new topic in itself.

If possible post links to recommended book lists by famous mathematicians working in that field

For instance the book list for algebraic topology by Hatcher is a good example


Thank you

  • $\begingroup$ I can recommend reading Chapter I in Robin Hartshorne's algebraic geometry, where one is introduced to the classical language of algebraic varieties. A lot of the commutative algebra machinery was developed because of Algebraic Geometry. If you want to learn more algebraic geometry after this, I highly recommend Foundations of Algebraic Geometry by Ravi vakil, which can be read for free here :math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf $\endgroup$ – Oliver E. Anderson Oct 15 '14 at 14:04
  • $\begingroup$ You can also start reading up on Algebraic Number theory, which is a nice follow up to Field Theory and Galois theory. You can take a look at Milne's notes here :jmilne.org/math/CourseNotes/ANT.pdf $\endgroup$ – Oliver E. Anderson Oct 15 '14 at 14:06

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