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An Introduction to Skolem's P-Adic Method for Solving Diophantine

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An Introduction to Skolem's P-Adic Method for Solving Diophantine An introduction to Skolem's p-adic method for solving Diophantine equations Josha Box July 3, 2014 Bachelor thesis Supervisor: dr. Sander Dahmen Korteweg-de Vries Instituut voor Wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam Abstract In this thesis, an introduction to Skolem's p-adic method for solving Diophantine equa- tions is given. The main theorems that are proven give explicit algorithms for computing bounds for the amount of integer solutions of special Diophantine equations of the kind f(x; y) = 1, where f 2 Z[x; y] is an irreducible form of degree 3 or 4 such that the ring of integers of the associated number field has one fundamental unit. In the first chapter, an introduction to algebraic number theory is presented, which includes Minkovski's theorem and Dirichlet's unit theorem. An introduction to p-adic numbers is given in the second chapter, ending with the proof of the p-adic Weierstrass preparation theorem. The theory of the first two chapters is then used to apply Skolem's method in Chapter 3. Title: An introduction to Skolem's p-adic method for solving Diophantine equations Author: Josha Box, [email protected], 10206140 Supervisor: dr. Sander Dahmen Second grader: Prof. dr. Jan de Boer Date: July 3, 2014 Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math 3 Contents Introduction 6 Motivations . .7 1 Number theory 8 1.1 Prerequisite knowledge . .8 1.2 Algebraic numbers . 10 1.3 Unique factorization . 15 1.4 A geometrical approach to number theory . 23 2 p-Adic numbers 32 2.1 The construction of the p-adic numbers . 32 2.2 From Qp to Cp ................................ 36 2.3 Topological properties of Qp and Cp ..................... 42 2.4 p-adic number fields . 43 2.5 Analysis on Cp ................................ 45 2.6 Newton Polygons . 48 3 Diophantine equations 54 3.1 Skolem's method for solving Thue equations . 54 3.2 Skolem's equations x3 + dy3 =1....................... 63 Reflection 66 Populaire samenvatting 68 Bibliography 70 5 Introduction One of the greatest recent achievements of a single mathematician may well have been Andrew Wiles' proof of Fermat's Last Theorem. In 1995 Wiles published his final paper, after having devoted seven years of his life to the problem. The theorem states that for any integer n > 2, there are no non-trivial integer solutions (x; y; z) to the equation xn + yn = zn: Of course, for n = 2 solving the Fermat equation is nothing else than finding Pythagorean triples, of which we know there exist infinitely many. How should one prove such a fact? In the case of finding infinitely many Pythagorean triples, it suffices to notice that (3; 4; 5) is a solution, which implies that (3n; 4n; 5n) is also a solution for every integer n ≥ 1. Wiles, however, had much more difficulty proving the non-existence of non-trivial solutions for n > 2. In general, if n is a natural number, A 2 Z and f 2 Z[x1; : : : ; xn], then an equation of the kind f(x1; : : : ; xn) = A for which integer solutions (x1; : : : ; xn) are searched, is called a Diophantine equation. In particular, Fermat's equation is Diophantine for every natural number n. The word Diophantine is derived from the Hellenistic mathematician Diophantus, who studied such equations in the third century AD. It was inside the margin of Diophantus' book Arithmetica that Pierre de Fermat scribbled his famous words \If an integer n is greater than 2, then xn + yn = zn has no solutions in non- zero integers x, y, and z. I have a truly marvelous proof of this proposition which this margin is too narrow to contain." Mathematicians have tried to find Fermat's \marvelous proof" ever since, without any success. Though Wiles did eventually prove the theorem in 1995, his proof uses techniques that could not have been known by Fermat in 1621. Therefore, most math- ematicians agree that Fermat had likely made a mistake in his proof. Avoiding the details, one could say that Wiles used a modern number theoretic approach to his class of Diophantine equations. Ever since Diophantus, and probably long before that as well, mathematicians have been occupied by Diophantine equations. In the past, this amounted to solving one such equation at a time. However, in the 20th century, new techniques allowed mathematicians to successfully solve entire classes of Diophantine equations. In this thesis one such technique, called Skolem's p-adic method, is investigated. This is done in Chapter 3, while the required prerequisite knowledge is studied in Chapters 6 1 and 2. In Chapter 1, an introduction to algebraic number theory is presented, while Chapter 2 is devoted to p-adic numbers. Algebraic number theory can be viewed as a foundation for most modern methods for solving Diophantine equations. Also, many recent methods, including Wiles' proof of Fermat's Last Theorem, use p-adic numbers. In Chapter 1, the prerequisites and introductory theory are presented in the first two sections. The latter sections focus on proving the unique decomposition of ideals of the ring of integers into prime ideals, Minkowski's theorem and Dirichlet's unit theorem. Furthermore, in Chapter 2 the p-adic numbers are defined and an algebraically closed complete extension is. The main result of this chapter is the p-adic Weierstrass prepa- ration theorem, of which Strassmann's theorem is found to be an immediate corollary. Together with Dirichlet's unit theorem, one could say that Strassmann's theorem is at the heart of Skolem's method described in Chapter 3. Motivations Parts of my own motivations for choosing the subjects of my thesis have been very well described by the German mathematician Richard Dedekind: \The greatest and most fruitful progress in mathematics and other sciences is through the creation and introduction of new concepts; those to which we are impelled by the frequent recurrence of compound phenomena which are only understood with great difficulty in the older view." In other words, I wanted to learn many new concepts and ideas. Therefore, I chose not to follow the shortest path towards solving Diophantine equations. Instead, I first dug deep into the algebraic and p-adic number theory, without continuously keeping their applications to Diophantine equations in mind. Section 2.3 is an example of a piece of theory that is not strictly necessary prerequisite knowledge for the methods applied in Chapter 3, while in my opinion the information contributes significantly to the understanding of the p-adic numbers. The idea of studying Skolem's p-adic method for solving Diophantine equations was proposed by my supervisor, who focuses on Diophantine equations in his own research as well. What fascinated me about Diophantine equations is that they are so accessible that I would be able to explain their meaning to, say, my grandmother, while mathematicians are often only able to say something about them using very advanced mathematics. To me it seemed a challenge to be able to understand such a sophisticated technique. Moreover, I had already enjoyed studying the first three chapters of Algebraic Number Theory and Fermat's Last Theorem by Stewart and Tall [16] in the honours extension of the course Algebra 2 and was therefore eager to learn more algebraic number theory as well. Furthermore, the theory of p-adic numbers was exactly one of those new concepts and ideas that Dedekind spoke of and it fascinated me that such an extraordinary idea had had such far-reaching implications. 7 1 Number theory In this chapter we will explore the number theory that is necessary for studying Diophan- tine equations in Chapter 3. Also, some examples of direct applications of the theory to suitable Diophantine equations are given in this chapter. Number theory is one of the oldest fields of mathematics and studies, in essence, the integers. The language in which the integers and its generalizations are studied is that of algebra. Therefore,the reader should have prerequisite knowledge of ring and field theory, group theory and some Galois theory. The theory as described in [4], [5] and [6] should be sufficient for understanding this chapter. 1.1 Prerequisite knowledge In this section, the important theory for understanding this chapter that the reader will be least likely to have come across is explained. Most of these facts are related to free abelian groups. We will frequently encounter such groups throughout this chapter. Definition 1.1. If G is an abelian group such that there exist g1; : : : ; gn 2 G that generate G and are linearly independent over Z, then G is called free abelian of rank n. The rank of a free abelian group is, similar to the dimension of a vector space in linear algebra, well-defined and any g 2 G can be expressed in a unique way as g = a1g1 + ::: + angn, where ai 2 Z for each i.A Z- linearly independent set generating G is also called a basis. Lemma 1.2. If G is free abelian of rank n with basis fx1; : : : ; xng and A = (aij) is an P n × n Z-matrix, then the elements yi = j aijxj form a basis for G if and only if A is unimodular, i.e. det A = ±1. T T Proof. Let x = (x1; : : : ; xn) and y = (y1; : : : ; yn) . Suppose that the yi form a basis. Then there exist matrices B and C such that x = By and y = Cx, so BC = In and the result follows. If A is unimodular, we can write ±A−1 = det(A) · A−1 = A~, the adjoint matrix of A which has integer coefficients.
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