Quadratic Reciprocity
Posted by John Baez
Next week is the last week of class here at UC Riverside! Yay!
I’m almost done teaching an undergraduate course on number theory, where the big final blaze of fireworks consists of proving Quadratic Reciprocity. I’m following a standard proof due to Eisenstein, outlined here:
- John Baez, Quadratic reciprocity: the big picture.
But, despite its pompous title, this outline doesn’t explain much. I don’t understand what makes Eisenstein’s proof tick, even after reading this play about it:
- Reinhard C. Laubenbacher and David J. Pengelley, Gauss, Eisenstein, and the “third” proof of the Quadratic Reciprocity Theorem: Ein kleines Schauspiel.
Maybe someone could explain what really makes this proof work?
In case you’re not in the know, Quadratic Reciprocity lets you tell if $p$ is the square of some integer mod $q$, assuming you know if $q$ is the square of some integer mod $p$. Here $p$ and $q$ are two odd primes.
So, it relates these two questions:
(A): Is $p$ is a square mod $q$?
(B): Is $q$ is a square mod $p$?
It asserts that
- If both $p$ and $q$ equal 3 mod 4, then questions (A) and (B) have opposite answers.
- Otherwise, questions (A) and (B) have the same answer.
People usually make this fact even prettier with a gadget called the Legendre symbol. This is written
$\left( \frac{p}{q} \right)$
and it equals $1$ if $p$ is a square mod $q$, and $-1$ otherwise. In terms of this trick, Quadratic Reciprocity says:
$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\left(\frac{p-1}{2}\right) \left( \frac{q-1}{2}\right)}$
It’s a fair amount of work to demonstrate this, at least using elementary methods. So, people have given many proofs. Gauss came up with 8. As of today, at least 227 proofs are known! They’re all listed here:
- Franz Lemmermeyer, Reciprocity Laws — book, website and chronological list.
I would like to understand how the elementry proof in my outline connects to more conceptual proofs, like those using algebraic number theory, which hint at generalizations such as Artin Reciprocity.
Re: Quadratic Reciprocity
John, I didn’t attempt to work through the details of the proof you sketched for your course, but have you looked at the postscript in John H. Conway’s The Sensual (Quadratic) Form? There seems to some sort of family resemblance between the elegant proof of quadratic reciprocity given by Conway and the Eisenstein proof, but you may find the Conway proof much more illuminating.
He starts off very sensibly by giving Zolotarev’s definition of the Jacobi symbol: If $a$ is prime to $n$, the symbol $(a/n)$ is defined to be the sign of the permutation of multiplying by $a$ on the set $\mathbb{Z}$ mod $n$. He comments, “This definition leads to an extremely simple proof of the quadratic reciprocity theorem. It is remarkable that this proof does not use either the notion of prime number, or even that of square number. We shall however use he fact that the sign of a permutation is multiplicative.”
Of course, one has to relate the Zolotarev definition of $(a/p)$ to the usual definition, but that’s not hard (just uses the fact that the multiplicative group of $\mathbb{Z}$ mod $p$ is cyclic).