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The study of reciprocity laws is a centerpiece of modern mathematics. Of the last ten Fields Medalists, two of them (Ngô Bảo Châu and Laurent Lafforgue) were awarded Fields Medals for their work on reciprocity laws. Andrew Wiles proved Fermat's Last Theorem by establishing a reciprocity law. Edward Witten works on reciprocity laws.

Not only are reciprocity laws popular today: they have a very distinguished history. Fermat, Euler, Lagrange, Legendre and Gauss were very interested in and spent a lot of time thinking about quadratic reciprocity. As is well known, Gauss called the theorem Aureum Theorema and was sufficiently motivated to understand it that he found eight proofs. The aforementioned mathematicians are not only celebrated for their work in number theory: they were also outstanding mathematician physicists and made outstanding advances in a number areas of math. (In Weil's history of number theory, he comments that while few mathematicians were interested in number theory in the early modern history of mathematics, those who were were of the highest quality.) So their strong appreciation of quadratic reciprocity is an indication that the phenomenon points toward some of the deepest and most interesting math.

Aside from being very fertile, quadratic reciprocity is in principle very easy to teach. The fact that

If $f(x) = x^2 -5$ then the prime divisors of members of the sequence $f(3)$, $f(4)$, $f(5)$, $f(6)$... are $2$, $5$, and those primes that have final digit $1$ or $9$

can be exhibited to middle schoolers. I would guess that people who have a solid understanding of two years of high school algebra can learn the full statement of quadratic reciprocity, its interpretation as a statement about which prime numbers factor further in quadratic number rings, its connection with cyclotomy and a hint as to how the phenomenon generalizes in $20$ hours or less. This statement needs qualification:

  • Here I don't mean understanding the proofs of all of the statements involved, but understanding the big picture using certain theorems from algebra and algebraic number theory as black boxes).

  • I would also emphasize that the material would have to be taught in a carefully constructed and streamlined fashion.

But with these qualifications, I think that my statement is true.

Despite all of this, very few math majors ever understand quadratic reciprocity. I would guess that the percentage that do is smaller than 1%. It's even uncommon for mathematicans to understand the theorem (I would guess that the percentage who understand is fewer than 50%, and maybe more like 20%). One reason for this is that most math majors aren't required to take a course in which they see quadratic reciprocity. Another reason for this is that courses in elementary number theory (where quadratic reciprocity is presented) don't present the theorem in a motivated way. In his lectures on The Practice of Mathematics (pg. 14 of the pdf) Robert Langlands (one of the major contributors to the study of reciprocity laws) wrote:

"I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law...appealing. I suppose, although I would not have – and could not have – expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl's book on the algebraic theory of numbers that I appreciated it as anything more."

I had the same initially reaction as Langlands did, and it was only four years after I first learned the statement of the theorem that I understood it.

What can we do as members of the mathematical community to raise awareness of reciprocity laws?

Can we change the algebra or elementary number theory course syllabi to address this issue? Can we push for the creation of a Nova or BBC series about reciprocity laws? Any other ideas?

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closed as off topic by Igor Pak, Ryan Budney, Chris Godsil, Leonid Positselski, Steven Landsburg Oct 28 '12 at 22:04

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Should intro courses be paying attention to research fads? This seems like more of an educational philosophy question rather than something specific to mathematics –  Ryan Budney Oct 28 '12 at 19:49
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Jonah, I suggest declaring May to be Quadratic Reciprocity Awareness Month (5 being the first prime congruent to 1 mod 4). You could organize a march on Washington DC, with sign-carrying algebraic number theorists bussed in from as far away as Cambridge, MA. It would probably get some news coverage, and I, for one, would be happy to participate. :-D –  Scott Aaronson Oct 28 '12 at 19:57
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Jonah, Quadratic reciprocity is already a big highlight in at least two summer math camps which many future mathematicians attend, namely the Ross Program at Ohio State University and the PROMYS program at Boston University. They even make t-shirts each summer with a proof on it, though there was a big scandal one summer when the store printing the t-shirts changed $(-1)^{\frac{p-1}{2}\frac{q-1}{2}}$ into $(-1)^{\frac{p-1}{2}}(-1)^{\frac{q-1}{2}}$ since it `looked better'. –  Patricia Hersh Oct 28 '12 at 20:08
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To start with, replace all commercials on TV by the lectures on reciprocity laws. Next, remove all the "Adult superstore" signs and replace them with reciprocity formulae on the highways. Next, make sure that nobody can get marijuana without first presenting 3 proofs of the quadratic reciprocity to the drug dealer. Continue in the same way, and the world will certainly improve at each step. Or, better, reincarnate on some other planet next time :). If you want me to be more down to Earth, create/update the relevant pages on Wikipedia so that they become readable. –  fedja Oct 28 '12 at 21:29
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Dear Henry, What I meant is simply that I am more and more upset by people closing a perfectly legitimate question, that attracted a handful of up votes and many comments in three hours after it was asked, moreover asked by a known mathoverflow user with some reputation. And I am even more upset that none of the closers did explain his reasons. I know that what follows should be on meta rather than here, but I see more and more legitimate questions to get closed. Perhaps there are too many people now with the right to close, and the minimal reputation needed to be allowed to do that... –  Joël Oct 29 '12 at 3:50
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