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2015 | Buch

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The Quadratic Reciprocity Law

A Collection of Classical Proofs

verfasst von: Oswald Baumgart

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is the English translation of Baumgart’s thesis on the early proofs of the quadratic reciprocity law (“Über das quadratische Reciprocitätsgesetz. Eine vergleichende Darstellung der Beweise”), first published in 1885. It is divided into two parts. The first part presents a very brief history of the development of number theory up to Legendre, as well as detailed descriptions of several early proofs of the quadratic reciprocity law. The second part highlights Baumgart’s comparisons of the principles behind these proofs. A current list of all known proofs of the quadratic reciprocity law, with complete references, is provided in the appendix.

This book will appeal to all readers interested in elementary number theory and the history of number theory.

Inhaltsverzeichnis

Frontmatter

Presentation of the Proofs of the Quadratic Reciprocity Law

Frontmatter

Chapter 1. From Fermat to Legendre

Abstract

After Bachet de Méziriac [1] had brought the theory of linear diophantine equations to a certain closure, mathematicians were faced with the question of solving equations of the second degree, in particular the binomial congruence of degree 2. In other words, the problem was to find simple conditions for the solvability of the congruence

$$\displaystyle{x^{2} \equiv p\bmod q,}$$

where p and q are given integers.

Oswald Baumgart

Chapter 2. Gauss’s Proof by Mathematical Induction

Abstract

1. Gauss distinguishes in his first proof, just as Legendre, eight different cases according to the different nature of the primes in question, so that the actual proof is seperated into eight proofs. The eight individual cases are:

If $q = 4n + 1$, $p = 4n + 1$ and $(\frac{p} {q}) = 1$, then we have to prove that $(\frac{q} {p}) = 1$;

If $q = 4n + 1$, $p = 4n + 3$ and $(\frac{p} {q}) = 1$, then we have to prove that $(\frac{q} {p}) = 1$;

If $q = 4n + 1$, $p = 4n + 1$ and $(\frac{p} {q}) = -1$, then we have to prove that $(\frac{q} {p}) = -1$;

If $q = 4n + 1$, $p = 4n + 3$ and $(\frac{p} {q}) = -1$, then we have to prove that $(\frac{q} {p}) = -1$;

If $q = 4n + 3$, $p = 4n + 3$ and $(\frac{p} {q}) = 1$, then we have to prove that $(\frac{q} {p}) = -1$;

If $q = 4n + 3$, $p = 4n + 1$ and $(\frac{p} {q}) = 1$, then we have to prove that $(\frac{q} {p}) = 1$;

If $q = 4n + 3$, $p = 4n + 3$ and $(\frac{p} {q}) = -1$, then we have to prove that $(\frac{q} {p}) = 1$;

If $q = 4n + 3$, $p = 4n + 1$ and $(\frac{p} {q}) = -1$, then we have to prove that $(\frac{q} {p}) = -1$.

Oswald Baumgart

Chapter 3. Proof by Reduction

Abstract

1. If q denotes a prime number, then $1,2,\ldots, \frac{q-1} {2}$ is a complete system of incongruent positive minimal¹ residues modulo q; on the other hand, if a is coprime to q, then a, 2a, …, $\frac{q-1} {2} a$ is a system of $\frac{q-1} {2}$ incongruent residues which do not necessarily form a half-system modulo q. If, in this last set, ρ ₁, …, ρ _λ are the positive and −σ ₁, …, −σ _μ the negative minimal residues modulo q, then we can observe that the ρ and σ are nonzero and pairwise distinct, hence congruent modulo q in some order to the numbers $1,2,\ldots, \frac{q-1} {2}$.

Oswald Baumgart

Chapter 4. Eisenstein’s Proof Using Complex Analysis

Abstract

1. Let p and q be two distinct positive odd primes and r the positive minimal residues modulo q; then we will have pr ≡ r ^′ or $\mathit{pr} \equiv -r^{{\prime}}\bmod q$, where r ^′ again denotes the positive minimal residues modulo q.¹ Thus we have

$$\displaystyle{\frac{\mathit{pr}} {q} = \frac{r^{{\prime}}} {q} + f\quad \text{or}\quad = -\frac{r^{{\prime}}} {q} + f,}$$

where f and f ^′ are integers. This implies

$$\displaystyle{\sin \Big(p\frac{2r\pi } {q} \Big) =\sin \frac{2r^{{\prime}}\pi } {q} \quad \text{or}\quad -\sin \frac{2r^{{\prime}}\pi } {q}.}$$

The properties of the sine function expressed by the above equation immediately leads to the result

$$\displaystyle{\mathit{pr} \equiv \frac{\sin \frac{2r^{{\prime}}p\pi } {q} } {\sin \frac{2r^{{\prime}}\pi } {q} } \bmod q,}$$

and this in turn shows that

$$\displaystyle{p^{\frac{q-1} {2} }\prod r \equiv \prod r^{{\prime}}\prod \frac{\sin \frac{2r^{{\prime}}p\pi } {q} } {\sin \frac{2r^{{\prime}}\pi } {q} } \bmod q,}$$

Oswald Baumgart

Chapter 5. Proofs Using Results from Cyclotomy

Abstract

1. Let ρ be a primitive root of the equation $\frac{x^{p-1}-1} {x-1} = 0$, where p is a positive odd prime, and let g be a primitive root modulo p; then we can order the roots of $\frac{x^{p-1}-1} {x-1} = 0$ in the following way:

$$\displaystyle{\rho,\rho ^{g^{2} },\rho ^{g^{4} },\ldots,\rho ^{g^{p-3} }\quad \text{and}\quad \rho ^{g},\rho ^{g^{3} },\ldots,\rho ^{g^{p-2} }.}$$

The expressions

$$\displaystyle{y_{1} =\rho ^{g} +\rho ^{g^{3} } +\ldots +\rho ^{g^{p-2} },\quad y_{2} =\rho +\rho ^{g^{2} } +\ldots +\rho ^{g^{p-3} }}$$

are called quadratic¹ periods of the cyclotomic equation $\frac{x^{p-1}-1} {x-1} = 0$. Using their property

$$\displaystyle{y_{1} - y_{2} = (\rho ^{-1}-\rho )(\rho ^{-3} -\rho ^{3})\cdots (\rho ^{-p+2} -\rho ^{p-2})}$$

and the relation

$$\displaystyle{(x -\rho ^{2})(x -\rho ^{4})\cdots (x -\rho ^{2(p-1)}) = x^{p-1} + x^{p-2} +\ldots +1}$$

we find

$$\displaystyle{(y_{1} - y_{2})^{2} = (-1)^{\frac{p-1} {2} }p.}$$

Now y ₁ + y ₂ = −1, hence we get

$$\displaystyle{y_{1}y_{2} = \frac{1 - (-1)^{\frac{p-1} {2} }p} {4}.}$$

Thus the two periods y ₁ and y ₂ are roots of the quadratic equation $f(x) = x^{2} + x + \frac{1} {4}(1 - (-1)^{\frac{p-1} {2} }p) = 0$.

Oswald Baumgart

Chapter 6. Proofs Based on the Theory of Quadratic Forms

Abstract

Recall that the complex of all equivalent forms of the same discriminant is called a form class. If the integers a, b, c in the form¹ (a, b, c) are coprime, then the form is called primitive. If the [greatest common] divisor σ of a, 2b, c is 1, then (a, b, c) is called a form of the first kind, and if σ = 2 it is called a form of the second kind. An ambiguous form² is a form in which the double middle coefficient 2b is divisible by the first. The form (1, 0, −D) is called the principal form of discriminant D; its class is called the principal class. If the outer coefficients of a form are positive, then the form itself is called positive.

Oswald Baumgart

Chapter 7. The Supplementary Laws of the Quadratic Reciprocity Law and the Generalized Reciprocity Law

Abstract

In our investigations we have made the assumption that we have already proved the supplementary laws of quadratic reciprocity, which can be expressed by the formulas

$$\displaystyle{(I)\ \ \Big(\frac{-1} {p} \Big) = (-1)^{\frac{p-1} {2} }\quad \text{and}\quad (\mathit{II})\ \ \Big(\frac{2} {p}\Big) = (-1)^{\frac{p^{2}-1} {8} }.}$$

In this section we will verify formulas (I) and (II) using the methods that we have already used in the chapters above for deriving the relation

$$\displaystyle{\Big(\frac{p} {q}\Big)\Big(\frac{q} {p}\Big) = (-1)^{\frac{p-1} {2} \cdot \frac{q-1} {2} }.}$$

First we remark that formula (I) is an immediate consequence of Fermat’s Theorem.

Oswald Baumgart

Chapter 8. Algorithms for Determining the Quadratic Character

Abstract

In the following we will present various ways of computing the symbol $(\frac{a} {b})$. Basically two methods have been applied. One is based on a direct application of the reciprocity law, and the other on a development of the fraction $\frac{a} {b}$ into a continued fraction. In the latter case $(\frac{a} {b})$ may be computed both from the quotients and the residues that occur in the development as a continued fraction. The first method is easily explained by an example. Assume we have to compute $(\frac{365} {847})$. Then we find

$$\displaystyle\begin{array}{rcl} \Big(\frac{365} {847}\Big)& =& \Big(\frac{847} {365}\Big)\qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{since}\ 365 \equiv 1\bmod 4, {}\\ & =& \Big(\frac{117} {365}\Big) =\Big (\frac{365} {117}\Big) =\Big ( \frac{14} {117}\Big) =\Big (\frac{117} {14} \Big) {}\\ & =& \Big( \frac{5} {14}\Big) =\Big (\frac{14} {5} \Big) =\Big (\frac{-1} {5} \Big) = +1, {}\\ \end{array}$$

hence¹ $(\frac{365} {847}) = +1$.

Oswald Baumgart

Comparative Presentation of the Principles on Which the Proofs of the Quadratic Reciprocity Law Are Based

Frontmatter

Chapter 9. Gauss’s Proof by Induction

Abstract

As we have already seen in Chap. 2, Gauss distinguishes eight cases in his first proof. This makes the first proof so long that it hardly can be found useful for the proof of such a simple law. Yet this lack of shortness is not so much a consequence of the principle of induction on which the proof is based but rather of the notation.

Oswald Baumgart

Chapter 10. Proofs by Reduction

Abstract

In Chap. 3 we have reproduced twelve proofs, which are all based on one and the same lemma that we will now briefly derive in its general form.

Oswald Baumgart

Chapter 11. Eisenstein’s Proofs Using Complex Analysis

Abstract

If r represents a half-system modulo q, then so does rp. Setting $\mathit{rp} \equiv \varepsilon r^{{\prime}}\bmod q$, where $\varepsilon = \pm 1$, and where r ^′ belongs to the same half-system as r, then for an arbitrary integer ω we have

$$\displaystyle{\frac{\mathit{pr}\omega } {q} \equiv \frac{\varepsilon r^{{\prime}}\omega } {q} \bmod \omega.}$$

This implies

$$\displaystyle{P\Big(\frac{\mathit{pr}\omega } {q} \Big) = P\Big(\frac{\varepsilon r^{{\prime}}\omega } {q} \Big),}$$

where P denotes any simply periodic function with period ω.

Oswald Baumgart

Chapter 12. Proofs Using Results from Cyclotomy

Abstract

In Chap. 5 we have collected the proofs that are based on theorems from cyclotomy. This theory was founded by Gauss when he was looking for another proof of his fundamental theorem. Already in 1796 [24] he announced the construction of the 17-gon. Apart from the fundamental theorems on imaginary numbers and functions, Gauss derived three (or, if you want, four) different proofs of the reciprocity law.

Oswald Baumgart

Chapter 13. Proofs Based on the Theory of Quadratic Forms

Abstract

1. The main idea in Gauss’s proof is, as Kummer observes, the fact that the number of actually existent genera is at most half the number of all possible genera. Gauss has shown that, if the reciprocity law was false, the number of actually existing genera had to be bigger than half the number of all possible genera. – Gauss distinguishes four different cases in his proof, which, as Dirichlet [12] has shown, can be reduced to two by choosing a suitable notation. Here too we have followed Dirichlet’s presentation.

Oswald Baumgart

Chapter 14. Final Comments

Abstract

In the following we would like to add a few historical remarks on the proofs listed in the First Part.

Oswald Baumgart

Chapter 15. Proofs of the Quadratic Reciprocity Law

Abstract

This appendix contains

(a)

A list of the known proofs of the quadratic reciprocity law; not included are proofs given in textbooks unless they contain a novel idea. Many of the proofs based on Gauss’s lemma as well as several others differ only marginally; nearly identical proofs were counted separately except when they were published by the same author.

(b)

A bibliography with references to the known proofs of the quadratic reciprocity law, together with a very brief description of the contents of the articles that were available to me.

Oswald Baumgart

Backmatter

Titel: The Quadratic Reciprocity Law
verfasst von: Oswald Baumgart
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-16283-6
Print ISBN: 978-3-319-16282-9
DOI: https://doi.org/10.1007/978-3-319-16283-6