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In my studies of physics and mathematics, I have encountered a fair bit of geometry, Lie group and representation theory, and real and complex analysis and I understand why these branches of mathematics are important. But I have learned very few applications of ring theory or other abstract algebra outside abstract algebra itself (save a few in number theory). At the same time, I believe it is considered vital for any aspiring mathematician to learn graduate level abstract algebra.

Why is abstract algebra considered to be so important? Examples of applications outside abstract algebra and outside mathematics would be appreciated.

To narrow the scope of the question down a bit, I am specifically asking about the theory of rings, fields, etc. I realize that the term 'abstract algebra' is a bit broader than what I intended.

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    $\begingroup$ And I thought that much of physics is applied group theory ... $\endgroup$ – Hagen von Eitzen May 17 '14 at 11:22
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    $\begingroup$ How do you study Lie groups and representation theory without abstract algebra? $\endgroup$ – Tobias Kildetoft May 17 '14 at 11:32
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    $\begingroup$ I guess it depends on what you mean by "deep". I am mainly familiar with this from the point of view of Lie algebras, where for example there is something like the Jantzen sum formula, which uses quite a bit of abstract algebra (a big part of the idea is that one can generalize a lot of things from fields to Dedekind domains). $\endgroup$ – Tobias Kildetoft May 17 '14 at 16:56
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    $\begingroup$ I guess algebraic quantum theory would be quite hard without algebra. $\endgroup$ – celtschk Aug 1 at 20:52
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    $\begingroup$ I won't write this as an answer, since most of it's over my head presently, but I've encountered quite a few references to abstract algebra in the field of cryptography, in privacy algorithms such as zero knowledge proofs. $\endgroup$ – user10478 Aug 8 at 16:18
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Abstract algebra has an interesting way of making a problem more transparent by forgetting about superfluous properties. I'll give some real world applications to illustrate:

Let's say you're a physicist studying the motion of a moving particle modeled by some differential equation. To find solutions to such an equation, one typically takes the Fourier transform and solves a corresponding algebraic problem. The Fourier transform is essentially allowing us to see past the complexity of arising from taking derivatives to illuminate an underlying algebraic problem.

As another example, lets say you are studying the effects of a gravitational field in a certain area of spacetime. Particles which travel through this area are subject to the curvature of spacetime induced by the gravitational field. Again this situation is extremely complicated. However, We can reduce the problem to an algebraic problem locally. That is, we use the tangent bundle and a smoothly varying metric to describe its motion.

Another example dates all the way back to Descartes. Geometric shapes are hard to understand, but imposing coordinates on such objects allows us to use algebraic techniques to understand the object better.

Anomolies in physics arise as elements of cohomology groups. Without the notion of a group, they would be very hard to calculate. You may not even know where to start.

The bottom line is, we know how to do algebra. It is the ability to translate of difficult problems into algebraic ones that makes it so useful.

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The fundamental particles "are" irreducible representations of the symmetry group of the universe.

If you want some applications of ring theory, it crops up in topology by way of cohomology rings.

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    $\begingroup$ This is a reason to study representation theory and Lie groups. I don't see how it motivates a fully general theory of rings or groups. $\endgroup$ – Jack M May 17 '14 at 11:32
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    $\begingroup$ I was answering "Why is abstract algebra considered to be so important? Examples of applications outside abstract algebra and outside mathematics would be appreciated." The author has asked ten questions in one. If (s)he wants only applications of, or morivation for, ring theory or more general group theory, the post should be revised to narrow the ridiculous scope of what I just quoted. My answer, in any case, offers an important example of an application "outside mathematics," unlike E.T.'s answer. $\endgroup$ – symplectomorphic May 17 '14 at 11:36
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Well, Abstract Algebra is a fundamental tool with many applications for natural sciences like Chemistry (Quantum Orbital Theory, etc.) and Physics (Quantum Mechanics, etc.) but this answer will focus on why it is important for Mathematics itself.

We all know that algebra is the language of Mathematics. It is the essential tool we use to notate our Mathematical ideas. In this case we can say that Abstract Algebra is the Linguistics of Mathematics. Every set of mathematical objects (real or complex numbers, points on elliptic curves etc.) talks a different language. Some of them are commutative under our operations (i.e. the 'verbs'), and some of them aren't. Let me give a better example. Let's think the set of polynomials with the property,

$$p(x) = 0$$

for $x = 1$. If you play with these polynomials soon or later you will realize that this set is closed under addition and scalar multiplication. Yes! You discovered the Vector Spaces now! As soon as you start to study and investigate their general properties, you will find out that the language of Vector Spaces is not only talked by the polynomials but also many other like $R^N$, $C^N$, etc. Since you already know the language they talk, even if you do know nothing about them, you will be knowing everything about them. This is the whole point of Abstract Algebra and it is really useful!

For sake of another example: If you know the language of groups (AKA group theory among mathematicians), you won't need to verify Euler's Totient Theorem since we already know Lagrange's Theorem holds for any finite group (in this case the finite group $U(n)$ closed under whatever). We already know everything about our mathematical objects even if we do know nothing about them.

Also why it is called abstract? Well, simply because for being able to talk about languages we must do an abstraction over mathematical objects and look for common properties between them.

This is my interpretation of the subject.

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