
#19
Mar2306, 11:00 AM

P: 355

i have come across this thing Reimann Zeta function once in my Complex number course...okay i try to see what was it..




#20
Mar2306, 11:05 AM

P: 355





#21
Mar2706, 11:10 PM

P: 38

To me it seems that the essence of abstract algebra is the study of structure for its own sake. While most of mathematics makes use of algebra in some way or another, the algebra is thought to represent specific, concrete objects and operations on those objects (i.e. integers, vectors, matrices, tensors, etc...).
However, abstract algebra studies the inherent structures of these operations on sets without regard for the particulars of representations or physical manifestations of such structures. As such, it allows us to focus on structure abstractly which sometimes leads to surprising results when we can establish correspondences between the structures of seemingly very different systems or representations. I would say perhaps the most fundamental concept of all in abstract algebra is that of ISOMORPHISM. It is often the case that certain systems or representations of systems reveal certain structures more clearly to our minds than others. For example, the complex numbers under multiplication exhibit the exact same structure as the group of scalings and rotations of vectors on the plane under composition. Complex numbers are easier to work with symbolically but scalings and rotations on the plane are easier to visualize. Establishing an isomorphism between these two systems allows us to freely move back and forth between them giving us the best of both worlds. For more intricate systems there could potentially be many useful representations or ways to visualize them. Abstract algebra allows us to not only move between representations  it also often allows us to construct entirely new representations to highlight certain aspects of a system's structure or to focus on specific substructures of interest. Also, by establishing general results for abstract structures we can immediately apply these results to all problems exhibiting similar structure rather than having to prove them for each system separately. 



#22
Mar2806, 12:15 PM

Sci Advisor
HW Helper
P: 9,421

i agree with what was just said, but not with the remarks that abstract algebra was conceived without applications in mind. this disagrees with what little i know about the origins of abstract methods in algebra by hilbert and his student emmy noether.
hilbert was trying to deal with concrete questions in invariant theory when he developed abstract methods of reasoning about polynomial rings. Noether extended this study to what are now called "noetherian rings", and applied abstract methods to study also homology groups of manifolds, possibly in an attempt to make more precise and invariant, the study of manifolds in topology. homology theory in topology had originally been a combinatorial theory that had trouble being proved a homeomorphism invariant. Apparently until Noether, homology groups were originally just numerical invariants, torsion numbers and ranks. abstract algebra also got a boost from emmy noether's father, max noether, who translated riemann's ideas on riemann surfaces in complex analysis into pure algebra, along with brill, to render the theory independent of some foundations of real analysis (dirichlet's principle) which had not yet been provided. hilbert also provided these by the way. van der waerden and emil artin and zariski and serre and grothendieck continued the development of abstract algebra to provide foundations for algebraic geometry, and render into algebra more tools from analysis, such as sheaf theory and its cohomology. serre's great paper GAGA is a monumental work translating analytic geometry into algebraic geometry and vice versa. grothendieck and others showed how to use algebra to translate over into algebraic geometry basic concepts of differential topology like chern classes, and provided an abstract generalized riemann roch theorem, first proved by hirzebruch on complex manifolds using cobordism theory from topology. abstract algebra methods also allow the study of singular spaces on the same footing with smooth spaces and manifolds, originally preferred in topology and differential geometry. this leads to a theory of degenerate varieties, a marriage of algebraic and topological singularity theory, and geometric compactifications of moduli spaces. so in my opinion abstract algebra was developed as a means to avoid the highly tedious computations in classical algebra that had become unmanageable, as well as questionable foundations in analysis. then it continued as a way to bring the power of calculus and analysis and topology to bear on algebra and number theory, and algebraic geometry, via sheaf cohomology. with the swing back in computing power, computational algebra is again in vogue. the foundations of analysis have also been rendered more sound, and algebra and analysis now cooperate ever more fully. 


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