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Nathan Jacobson's Basic Algebra I, II covers many topics in Algebra that is probably even beyond many pure mathematics full professor's scope of knowledge, unless the professor is specialised in algebra.

My question is, if the following topics are included in Basic Algebra:

  1. Modules over a PID;
  2. Galois Theory;
  3. Algebras over a Field;
  4. Lattices and Boolean Algebras.
  5. Classical Representation Theory;
  6. Homological Algebra;
  7. Commutative Ideal Theory;
  8. Field Theory;
  9. Valuation Theory;
  10. Dedekind Domains.

Then what is considered (according to N. Jacobson's standard) as "Intermediate Algebra" and what is considered "Advanced Algebra"? Any list of topics and recommended textbooks on Intermediate Algebra and Advanced Algebra?

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    $\begingroup$ Probably cutting-edge academic articles on abstract algebra or books on specific sub-fields. Clearly the usage is different from high-school/remedial level basic/intermediate/advanced algebra. Probably he would not recognize "intermediate" as any kind of useful qualification for graduate-level studies like these. $\endgroup$ – Daniel R. Collins Jan 25 at 23:56
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    $\begingroup$ Jacobson probably means "basic" in the sense of "the basics for a researcher in the field". Those are all topics I would consider basic knowledge for anyone doing research in algebra (with the possible exception of lattices and Boolean algebras, which aren't as big of areas of research as they used to be). $\endgroup$ – Daniel Hast Jan 26 at 4:53
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    $\begingroup$ Basic Algebra I-II are great books. The intended reader is an advanced undergrad or a first year grad student. Depending on how much ivy your school has. They seek to form a kind of a common ground to a topics in algebra course, but some parts of vol. II could serve as such topics themselves. Anyway, basic here means roughly distant from current research, but something a trained mathematician may need to know. All this stuff was finalized well before world war II (the early chapters date to 1800s). You should not read too much into Basic here. $\endgroup$ – Jyrki Lahtonen Feb 9 at 22:48

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