http://www.design-ring.com/groups-rings-and-fields/
Practical applications of abstract algebra?
I think abstract algebra (groups, rings, fields, modules, etc.) is really cool and actually quite fun, but I’m having a hard time grasping (or seeing, I guess) the practical side of it. What sort of real-world problems are these structures used to solve? Can anyone provide specific examples or recommend some books that might discuss practical examples? I know that medical imaging techniques are based around carefully constructed mathematical models that involve some pretty advanced real analysis; is there anything like that for rings, modules, and the like?
I know, or at least *think* I know, that Galois theory is all about providing solutions to polynomial equations (or showing, somehow, that there aren’t any) by more or less assigning to each polynomial some group, called the Galois group, and using the properties of that group to discuss if there are, or aren’t, solutions. I can see immediately why that’s relevant and useful. But there’s got to be other things…
I do not really know but I recall when I was at university, I picked up a physics book and there was a lot of group theory being used, so I know that physics has found applications.
I found this interesting answer:
http://answers.yahoo.com/question/index?qid=20060730062321AAaZdy5
And this interesting passage from wikepedia:
* An understanding of group theory is also important in physics and chemistry and material science. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the “possible” physical theories. Examples of the use of groups in physics include: Standard Model, Gauge theory, Lorentz group, Poincaré group
* In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and to construct molecular orbitals.
* Group theory is used extensively in public-key cryptography. In Elliptic-Curve Cryptography, very large groups of prime order are constructed by defining elliptic curves over finite fields.
Additional links:
http://mysite.du.edu/~jcalvert/phys/groups.htm
http://www.answers.com/topic/group-theory?cat=technology
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