Quadratic Reciprocity I

We now come to the most important result in our course: the law of quadratic reciprocity, or, as Gauss called it, the aureum theorema (“golden theorem”). Many beginning students of number theory have a hard time appreciating this golden theorem. I find this quite understandable, as many first courses do not properly prepare for the result by discussing enough of the earlier work which makes quadratic reciprocity an inevitable discovery and its proof a cause for celebration. Happily, our study of quadratic rings and the quadratic form x − Dy has provided excellent motivation. There are also other motivations, involving (what we call here) the direct and inverse problems regarding the Legendre symbol. A faithful historical description of the QR law is especially complicated and will not be attempted here; we confine ourselves to the following remarks. The first traces of QR can be found in Fermat’s Lemma that −1 is a square modulo an odd prime p iff p ≡ 1 (mod 4), so date back to the mid 1600’s. Euler was the first to make conjectures equivalent to the QR law, in 1744. He was unable to prove most of his conjectures despite a steady effort over a period of about 40 years. AdrienMarie Legendre was the first to make a serious attempt at a proof of the QR law, in the late 1700’s. His proofs are incomplete but contain much valuable mathematics. He also introduced the Legendre symbol in 1798, which as we will see, is a magical piece of notation with advantages akin to Leibniz’s dx in the study of differential calculus and its generalizations. Karl Friedrich Gauss gave the first complete proof of the QR law in 1797, at the age of 19(!). His argument used mathematical induction(!!). The proof appears in his groundbreaking work Disquisitiones Arithmeticae which was written in 1798 and first published in 1801. The circle of ideas surrounding quadratic reciprocity is so rich that I have found it difficult to “linearize” it into one written presentation. (In any classroom presentation I have found it useful to begin each class on the subject with an inscription of the QR Law on a side board.) In the present notes, the ordering is as follows. In §1 we give a statement of the quadratic reciprocity law and its two supplements in elementary language. Then in §2 we discuss the Legendre symbol, restate QR in terms of it, and discuss (with proof) some algebraic properties of the Legendre symbol which are so important that they should be considered part of the quadratic reciprocity package. In §3 we return to our “unfinished theorems” about representation of primes by |x −Dy| when Z[ √ D] is a PID: using quadratic reciprocity, we can state and prove three bonus theorems which complement Fermat’s Two Squares Theorem. In §4 we define and discuss the “direct and inverse problems” for the Legendre symbol and show how quadratic reciprocity is useful for both of these, in particular for rapid computation of Legendre symbols. More precisely, the computation would be rapid if we could somehow avoid having to factor numbers