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I recently posted a rather soft question on MSE, (Why algebraic closures?) and the post attracted relatively a lot of votes, and two answers which are interesting.
I do feel, however, that I am unsatisfied by these answers. On the other hand, I am also not sure what answers I am looking for... I was wondering whether or not it would seem reasonable to post this into MO (despite not knowing what answer I am looking for)
I'm a bit surprised that no one mentioned the question of finding eigenvalues of square matrices. If your field is algebraically closed, you can diagonalize almost all matrices. [Correction: I see now that Qiaochu did in fact bury it in the middle of a paragraph.]
Another killer application is Hilbert's Nullstellensatz. You could ask why algebraic geometry is useful while sine geometry isn't.
Scott, while I do wonder sometimes why algebraic geometry is useful, I think that you are somewhat missing the point. In this aspect Bill's answer was slightly closer to what I was looking for.
Clearly for linear algebra (and its advanced derivatives) the algebraic closure is very useful. But we're no longer at the stage that most things are directly related to linear algebra. Maybe it is time to consider other forms of closures? I'm not sure exactly what I am looking for, but I think that I a pretty sure that I am not looking for the answer "because those are very useful in AG, and AG is very useful!"...
(I suppose I am making slightly exaggerated, it's just that it's so hot today...)
@Scott: I am not familiar with the term "sine geometry". Can you point me to a reference? Thanks.
@Will"y": I think that Scott was referring to geometry related to sine functions over a sine-closed field (much like algebraic operations such as matrices often take place over an algebraically closed field)
@Asaf: ah I see, it is in reference to your MSE post.