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Are there any research/books about quadratic residue in other number system.

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 The downvote is not from me. But, your question is a bit vague. Perhaps, you could expand it by stating something on your background and/or what you need the book for. Moreover, if the intent is more introductory, you might be better served asking it at math.stackexchange.com – quid Aug 18 2011 at 23:11
Why are people voting to close this? There is tons of classical work on this; see en.wikipedia.org/wiki/… I'm not sure what book to recommend, but I imagine other readers of MO will. – David Speyer Aug 18 2011 at 23:38
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 The book by Mike Rosen "Number Theory in Function Fields" discussed quadratic reciprocity in F_q[x] for odd q. More generally you can use the Hilbert symbol to formulate a quadratic reciprocity law in all number fields and function fields over finite fields (function fields of characteristic 2 may need a separate treatment depending on your point of view). I agree with quid that you should explain why you are asking this question. – KConrad Aug 18 2011 at 23:58
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 D. Hilbert: "Ueber die Theorie des relativquadratischen Zalhkörpers," Math. Annalen 51 (1899), 1-127. – Felipe Voloch Aug 19 2011 at 11:10
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 I'm voting to close because it's much too vague (what exactly does OP mean by "other number system(s)"?) and has the appearance of a fishing expedition. I hope OP will edit it into something better and, as quid indicates, if it's not of interest to research mathematicians (see the faq) then post it to math.stackexchange. – Gerry Myerson Aug 19 2011 at 12:58

closed as off topic by Gjergji Zaimi, David Roberts, Felipe Voloch, Gerry Myerson, Igor Rivin Aug 19 2011 at 15:20

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