Gauss Sums, Quadratic Reciprocity, and the Jacobi Symbol

Abstract

Our first goal in this chapter is to present Gauss’s sixth proof of his Law of Quadratic Reciprocity. The presentation here follows [32, §3.3] fairly closely, except that our Gauss sums are over the complex numbers, as opposed to ibid. where Gauss sums are considered over a finite field. Later in the chapter we introduce the Jacobi symbol and study its basic properties. The Jacobi symbol will make an appearance in Chapter when we give a proof of the Three Squares Theorem.

Appendices

Exercises

• 7.1 Compute $\tau _p$ for $p=3, 5$, and verify Lemma 7.1 directly.

• 7.2 ($\maltese$) Compute $\tau _p$ for $p=17$.

• 7.3 Show that the complex number C defined in Equation (7.1) is an algebraic integer.

• 7.4 Prove the second part of Lemma 7.4.

• 7.5 Prove the second part of Theorem 7.3.

• 7.6 Determine all natural numbers n such that $(n/15)=+1$.

• 7.7 Determine (215 / 997) and (113 / 1093) using the Jacobi symbol.

• 7.8 Find five pairs of integers (a, b) such that the Jacobi symbol $(a/b) = +1$ but $x^2 \equiv a \ \mathrm {mod}\ b$ is not solvable.

• 7.9 Show that for all $n > 1$ we have the following identities for Jacobi symbols

  $$ \left( \frac{n}{4n-1}\right) = - \left( \frac{-n}{4n-1}\right) = 1. $$

• 7.10 Show that for an integer d with $|d|>1$ we have

  $$ \left( \frac{d}{|d|-1}\right) = {\left\{ \begin{array}{ll} 1 &{} d > 0;\\ -1 &{} d < 0. \end{array}\right. } $$

• 7.11 Let $k \in \mathbb N$, and let $\gcd (d, k) =1$. Prove that the number of solutions of $x^2 \equiv d \ \mathrm {mod}\ 4 k$ is

  $$ 2 \sum _{f \mid k \atop f \text { squarefree}} \left( \frac{d}{f}\right) . $$

• 7.12 Show that for an odd prime p, and $a \in \mathbb N$ with $p \not \mid a$, we have

  $$ \left( \frac{a}{p}\right) = \left( \frac{a}{p-4a}\right) . $$

• 7.13 This exercise gives another proof of the Law of Quadratic Reciprocity due to Rousseau [94]. The proof uses a bit of group theory. Let p, q be odd primes, and define $G = (\mathbb Z/pq\mathbb Z)^\times / \{ \pm 1\}$.

  • a. Show that the set

    $$ S =\left\{ (x, y) \mid 1 \le x \le p-1, 1 \le y \le \frac{q-1}{2}\right\} $$

    is a set of representatives for G. What is the product of elements of S modulo $\{\pm 1\}$?

  • b. Show that the set

    $$ S' =\left\{ (z \ \mathrm {mod}\ p , z \ \mathrm {mod}\ q ) \mid 1 \le z \le \frac{pq-1}{2}\right\} $$

    is another set of representatives of G. Determine the product of elements of $S'$ modulo $\{\pm 1\}$.

  • c. Derive the Law of Quadratic Reciprocity from the first two parts.

Notes

1.1 Proofs of quadratic reciprocity

As mentioned in Chapter 6, the Law of Quadratic Reciprocity was conjectured by Euler around 1745, in a paper titled “Theoremata circa divisores numerorum in hac forma $pa^2 \pm qb^2$ contentorum” available from the Euler Archive at

http://eulerarchive.maa.org/index.html

though here the conjecture is not explicitly stated as such. The explicit formulation of the conjecture appears in a later paper of Euler’s, titled “Observationes circa divisionem quadratorum per numeros primos” available at

http://eulerarchive.maa.org/pages/E552.html

Gauss noted in his notebook that he had found a proof on April 8, 1796. So far over 200 proofs of the Law of Quadratic Reciprocity have been obtained by various mathematicians. Franz Lemmermeyer, the author of [32], maintains a website that keeps track of the various proofs of theorem. The website is available at

http://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html

1.2 Generalizations

One can generalize the Law of Quadratic Reciprocity in two different directions, one is by considering higher powers, and the other by considering other number fields, introduced in the Notes to Chapter 5. For introductions to reciprocity laws for higher powers we refer the reader to Lemmermeyer [32] or Cox [14], especially §4. For the generalization of Quadratic Reciprocity to other number fields, known as Hilbert’s Law of Reciprocity, see the Notes to Chapter 8.