Thread: beyond abstract algebra

Does anybody know what topic lies beyond or higher than abstract algebra?

"Abstract Algebra" is a deliberately scary name intended to weed out those not truly serious about mathematics. This is why it is a weeding course. If you can't hack it, please rethink your career goals! After that, it's just linear algebra or maybe even just algebra. Hang in there long enough and you may start talking about "an" algebra.

Significantly Tongue-in-Cheek :-)

Seriously, it has been impossible to "order" mathematics since about 1900 at the very latest. Various branches grow, seemingly separated, until they absorb each other. You DO have to get used to it. As great as is the depth of mathematics, its breadth may be even greater.

abstract algebra is a vast field, which is usually just skimmed in a course of the same name. for example, one can study ring theory in some detail (as there is an amazing diversity of rings), or modules, or algebras. there is a well-developed theory of real-closed fields, and a lot of linear algebra can be generalized to division rings.

moreover, the methods of abstract algebra are then used in algebraic topology, number theory, homological algebra, category theory, and differential geometry, to name just a few subjects. and these are fairly "broad areas", within each one, are a number of interesting and complex subjects in their own right. and many of these subjects "interleave", so it's hard to say exactly "which" subject you're actually studying.