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Post Closed as "off topic" by Igor Pak, Ryan Budney, Chris Godsil, Leonid Positselski, Steven Landsburg
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# What can we do to raise awareness of reciprocity laws?

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?