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What are some interesting applications of the concept of homomorphism?

Example: If there is a homorphism from a ring $R$ to a ring $r$ then a solution to a polynomial equation in $R$ gives rise to a solution in $r$. e.g. if $f:R \rightarrow r$ and $X^2+Y^2=0$ then $f(X^2+Y^2)=f(0), f(X^2)+f(Y^2)=0, f(X)^2+f(Y)^2=0, x^2+y^2=0$

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    meta discussion here: http://meta.math.stackexchange.com/questions/765/are-we-opposed-a-priori-to-fishing-expedition-questions-like-this-one2010-09-03
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    @user1613: I would appreciate if you could be much more specific about what you want. Are you a beginning student of abstract algebra who wants to understand why people care about homomorphisms? Are you a first-time teacher who wants good examples for his/her students? Etc.2010-09-03
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    @user1613: In fact homomorphism is a widely concept even outside the mathematical subject we usually call *abstract algebra*. However, it is always a mapping between to structured objects of the same kind that and that map is structure preserving. By structure I mean an operation.2010-09-03
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    @Qiaochu Yuan: I studied some algebra years ago, mostly on the basis of memorizing stuff to pass exams. It seemed on the whole to be a lot of abstract nonsense. In particular I was nonplussed by homomorphisms. It wasn't a concept that felt nice and good. It gave more a feeling of discomfort. Then the other day I wrote the question about arithmetic of n-dimensional arrays and the above example about polynomial solutions occured to me and suddenly the concept of homomorphism felt good so I guess I'm looking for more examples that improve the taste of the concept of homomorphisms. more...2010-09-03
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    ... Something that adds some salt or spice. A sugar-coated pill to attach a Pavlovian feel-good factor to the concept. The more concrete the better.2010-09-03
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    @user1613: you should mention all this in the question.2010-09-03

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