Where does Quadratic Reciprocity point?

In his book Lectures on the theory of algebraic numbers, Hecke says that the content of the quadratic reciprocity theorem, formulated and proved entirely in terms of rationals (integers) points beyond the domain of rational numbers.

He is talking about the algebraic numbers but how is it seen that it points elsewhere?

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Not all algebraic numbers are rational. –  Joshua Shane Liberman Feb 3 '11 at 17:57
What is the context of this statement, by the way? If you are really interested in what specifically Hecke meant it would be helpful to have a reference. –  Qiaochu Yuan Feb 3 '11 at 19:35
That way: $\longrightarrow$ –  Mariano Suárez-Alvarez Feb 3 '11 at 20:18
$\longleftarrow$ perhaps? –  uncle brad Feb 3 '11 at 22:48

2 Answers

Quadratic reciprocity suggests that there ought to be a cubic or quartic reciprocity law. There are, but even to state them cleanly (let alone prove them) it is best to work in $\mathbb{Z}[\omega]$ or $\mathbb{Z}[i]$ respectively instead of the integers, roughly speaking because of Kummer theory. See, for example, the Wikipedia articles, and in particular the following quotes from Gauss on quartic reciprocity:

The theorems on biquadratic residues gleam with the greatest simplicity and genuine beauty only when the field of arithmetic is extended to imaginary numbers, so that without restriction, the numbers of the form a + bi constitute the object of study ... we call such numbers integral complex numbers.

and cubic reciprocity:

The theory of cubic residues must be based in a similar way on a consideration of numbers of the form a + bh where h is an imaginary root of the equation h3 = 1 ... and similarly the theory of residues of higher powers leads to the introduction of other imaginary quantities.

The proof of quadratic reciprocity by Gauss sums also naturally leads to an elegant proof using the Galois theory of number fields which generalizes to the Kronecker-Weber theorem and suggests the beginnings of class field theory.

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Well said, but did Hecke mean that? –  lhf Feb 3 '11 at 19:20
Why would you expect a cubic reciprocity law? Is there any reason other than wishful thinking. –  quanta Feb 3 '11 at 19:45
@lhf: Yes, there is every reason to think that Hecke meant this. The investigation of higher reciprocity laws has been one of the major thrust of number theory from the publication of Gauss's Disquitiones through to the contemporary era, and Hecke was very much part of this tradition. –  Matt E Feb 3 '11 at 20:05
@quanta: considerations of mathematical beauty suggest that squares are probably not special, just easier. In any case, musings about cubic reciprocity go back to Euler, so it's not exactly a weird idea. In fact, it seems a little weird to me to see quadratic reciprocity and not wonder "wow! I wonder if this generalizes." Doubtless Euler and Gauss also did countless calculations which enabled them to make such conjectures. –  Qiaochu Yuan Feb 3 '11 at 20:37
@quanta: like I said, Euler and Gauss did plenty of calculations before making any such conjectures. They were very familiar with the evidence, so to speak. –  Qiaochu Yuan Feb 5 '11 at 16:47

Just to elaborate on Qiaochu's answer: one way to prove quadratic reciprocity is to observe that, for an odd prime $p$, the quadratic field $\mathbb Q(\sqrt{\pm p})$ (the sign being chosen so that $\pm p \equiv 1 \bmod 4$) is contained in $\mathbb Q(\zeta_p)$ (the field obtained by adjoining a primitive $p$th root of unity to $\mathbb Q$), as can be seen by using Gauss sums, and combining this with the irreducibility of the $p$th cyclotomic polynomial (which shows that $Gal(\mathbb Q(\zeta_p)/\mathbb Q) = (\mathbb Z/p)^{\times}$).

All these concepts go back to Gauss's Disquitiones Arithmeticae, which served to ispire and guide all the subsequent developments in number theory in the 19th century.

Gauss himself introduced his Gaussian integers (i.e. the ring $\mathbb Z[i]$) as part of his investigations of biquadratic (i.e. fourth power) reciprocity. His student Eisenstein investigated cubic reciprocity (and introduced the ring $\mathbb Z[\zeta_3]$ as a tool to this end).

Later in the 19th century Kummer investigated higher prime power reciprocity laws, and was led to the invention of the main concepts of algebraic number theory (ideals, unique factorization into prime ideas, the class group and class number, all in the context of the fields $\mathbb Q(\zeta_p)$) as part of his investigation.

The investigation of higher reciprocity laws continued. When the class group of $\mathbb Q(\zeta_p)$ is non-trivial, especially when it has order divisible by $p$, new phenomena emerged, which led Hilbert to the concept of Hilbert class field.

Out of all this the general conception of class field theory emerged, and was finally established by Takagi in the early 20th century.

Hecke was aware of all this tradition, and it to this tradition and these developments that he is referring in his remark.

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