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I've taken an undergraduate course in ring and group theroy, but haven't studied number theory formally. I've noticed many important results in number theory have been generalized in group/ring theory (e.g. Lagrange Thm generalizing Fermats Little Thereom).

My question is are there any results in elementary number theory that have not been generalized using elementary group and ring theory?

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I don't know that this is a great question as written. The narrowest reading is "are there results in number theory which hold for Z and which don't hold for other rings" and this seems to be a difficult request to fulfill. Do you actually mean something broader than this? –  Qiaochu Yuan Aug 16 '10 at 18:04
    
To re-pharse my question: Are there results in number theory which hold for Z and which don't hold for other rings? –  Digital Gal Aug 16 '10 at 18:35
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Quadratic reciprocity does not generalize to all groups or rings. It is a genuinely arithmetic statement and can be extended to Z[i] and other similar extensions of Z, but not to every commutative ring. What separates number theory from abstract algebra is any result that relies on the finiteness of the quotient rings Z/n (e.g., this lies behind the defn of the zeta-function). –  KCd Aug 16 '10 at 23:28
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up vote 7 down vote accepted

Problems for integers whose generalizations in algebraic number theory have not been developed (or at least considered) are rare. The only examples I know of are:

  1. Ancient problems without a strong relation to modern algebraic techniques. This includes the existence of odd perfect numbers, and compositeness of Fermat numbers $2^{2^k}+1$. Keep in mind, though, that most of the very old problems have been understood or finished off by modern techniques, such as the congruent number problem and Fermat's Last Theorem being reduced to facts about elliptic curves, or the probabilistic approach to the distribution of special types of primes (twin, Mersenne, k-term progressions, etc), and these better-understood problems are more amenable to generalization beyond the integers.

  2. Questions of combinatorial nature, such as covering congruences or arithmetic progressions (van der Waerden theorems). Here the problems can be generalized to other rings, but most interest has focused on the integer case.

  3. Uses of number theory in logic and computer science, such as Goedel coding, complexity of Presburger arithmetic, or other problems where iterated exponentiation appears. Algebraic number theory is primarily about problems described by polynomial equations, and the reduction of exponential Diophantine equations to polynomial ones (as in the solution of Hilbert's 10th problem) is not direct enough to allow the use of the standard algebraic tools.

For everything else, where standard algebraic techniques or analytic ones (zeta functions, Diophantine approximation, sieves, transcendence theory, complex analysis, etc) are applicable to a problem over the integers, there has usually been an effort to search for analogues of those methods in algebraic number theory, p-adic number theory, geometry over finite fields, and complex geometry. This applies to all the major techniques and theories developed since 1800. So to find examples one needs to move away from "central core" number theory and look at questions that are not, so far, primarily studied by the methods of algebraic number theory and algebraic geometry.

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I contend that every result that could reasonably be called "elementary" number theory also holds for $\mathbb{F}_q[x]$, the ring of polynomials in one variable over a finite field. Quadratic reciprocity fits the bill here, for example. This is the subject of at least one book.

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Vinogradov's or Chen's works on the Twin prime conjecture and Goldbach conjecture can be argued to be elementary number theory. Do they have function field analogues? –  user1119 Aug 17 '10 at 17:03
    
@George S.: that may have been a rhetorical question but I think it would be very interesting to come up with a reasonable function field analogue of either the twin prime conjecture or Goldbach and it is not clear to me that none exists. –  Qiaochu Yuan Aug 17 '10 at 18:19
    
Yes, but it is not clear that they exist, either. Your claim would need further substantiation than is given at the moment. –  user1119 Aug 17 '10 at 19:01
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For an analogue of Goldbach, look at G. Effinger, D. Hayes, "Additive number theory of polynomials over a finite field," Oxford Univ. Press, New York, 1991. For twin primes, look at links on Effinger's web page skidmore.edu/~effinger/webvita.htm. –  KCd Aug 17 '10 at 21:34
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