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What are the different parts in Abstract Algebra? For example Algebraic Geometry, Algebraic Number theory, Universal algebra, Category theory and so on, can all these subjects be categorized? I'm a second year math student, and I need to do a presentation for other second year student, and I need to tell something about algebra and which different fields there are in abstract algebra. Because I haven't studied any of those fields that I read, I find it hard to see how you could divide all the parts of algebra.

I'm trying to let it make sense to myself. I would say that in abstract algebra, you study algebraic structures like groups, rings, algebra's. And if I inteprete @Matt Pressland correctly, you could use those algebraic structures to solve problems in geometry, number theory and topology etc. I don't understand how representation theory, Universal algebra, Category theory fits in this picture.

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A definitive list is likely to be impossible, and there are many overlaps... But you should probably mention representation theory. Algebraic geometry (and I suppose algebraic number theory) are in some sense (but not all senses) applications of abstract algebra, rather than being subfields. –  Matt Pressland Nov 19 '13 at 16:46
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can they be categorified? I'm not sure. You should consider the stable category formed by modding out by projective algebra. Category theory is the picture; your life is just a object and love is the functor. –  Peter Halburt Nov 19 '13 at 16:54
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Don't forget homological algebra. –  Tyler Holden Nov 19 '13 at 16:55
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You might take a look at en.wikipedia.org/wiki/List_of_algebras and are you familiar with the picture intothecontinuum.tumblr.com/post/4073240202/… from arxiv.org/abs/gr-qc/9704009? –  isomorphismes Nov 19 '13 at 17:04
    
Here is an example of how category theory can show a different perspective on something else from algebra: a group is a category with one object. (All the automorphisms are implied.) –  isomorphismes Nov 21 '13 at 6:25

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