Abstract Algebra: Beyond and Higher?

In summary, There are many fields of abstract mathematics, but they are not necessarily ranked in terms of being "beyond and higher" than each other. Some potential fields that could be considered beyond abstract algebra include Category Theory, Representation Theory, Algebraic Geometry, and Set Theory.
  • #1
alvin51015
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0
I was wondering if there is a field of mathematics which lies beyond and higher than abstract algebra? If it exists could someone tell me the name of that field? Thanks.
 
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  • #3
There are many fields of abstract mathematics. They are generally not "ordered" so that we cannot think of one as being "beyond and higher" than another. But I think perhaps "Category Theory" (http://en.wikipedia.org/wiki/Category_theory) might qualify here.
 
  • #4
Representation theory probably counts.
 
  • #5
Algebraic geometry should count
 
  • #6
alvin51015 said:
I was wondering if there is a field of mathematics which lies beyond and higher than abstract algebra? If it exists could someone tell me the name of that field? Thanks.
I would say Set Theory. Set theorists keep adding axioms to their Set theory and expect us to do math(such as category theory, mathematical analysis and pretty much all of the known math) withing their formal Axiomatic Set Theory.
 

Related to Abstract Algebra: Beyond and Higher?

1. What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures, such as groups, rings, and fields, and their properties. Unlike elementary algebra, which deals with numbers and operations, abstract algebra deals with abstract mathematical objects and their relationships.

2. What are some applications of abstract algebra?

Abstract algebra has many applications in different fields of mathematics, such as number theory, geometry, and topology. It is also widely used in computer science, physics, and engineering for solving complex problems and developing algorithms. Some examples of applications include cryptography, coding theory, and error-correcting codes.

3. What are some key concepts in abstract algebra?

Some key concepts in abstract algebra include groups, which represent symmetry and transformation; rings, which generalize the concept of numbers and arithmetic; and fields, which extend the concept of numbers and allow for division. Other important concepts include homomorphisms, isomorphisms, and substructures.

4. How does abstract algebra differ from other branches of mathematics?

Abstract algebra differs from other branches of mathematics in that it focuses on the abstract properties and structures of mathematical objects, rather than their specific values. It also uses a more rigorous and axiomatic approach, where the rules and properties of algebraic structures are defined and studied systematically.

5. Is abstract algebra difficult to learn?

Abstract algebra can be challenging to learn, as it requires a solid understanding of mathematical concepts such as sets, functions, and proofs. However, with patience and practice, anyone can develop a strong understanding of abstract algebra. It is helpful to have a strong foundation in algebra and other areas of mathematics, but it is not necessary to be an expert in those subjects.

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