An Exposition of the Eisenstein Integers
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The Class Number One Problem for Imaginary Quadratic Fields
MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM JEREMY BOOHER Gauss found 9 imaginary quadratic fields with class number one, and in the early 19th century conjectured he had found all of them. It turns out he was correct, but it took until the mid 20th century to prove this. Theorem 1. Let K be an imaginary quadratic field whose ring of integers has class number one. Then K is one of p p p p p p p p Q(i); Q( −2); Q( −3); Q( −7); Q( −11); Q( −19); Q( −43); Q( −67); Q( −163): There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner's paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner's proof was essentially correct and in fact equiv- alent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new approach based on modular curve, reducing the class number + one problem to finding special points on the modular curve Xns(n). For certain values of n, it is feasible to find all of these points. He remarks that when \N = 24 An elliptic curve is obtained. This is the level considered in effect by Heegner." Serre says nothing more, and later writers only repeat this comment. This essay will present Heegner's argument, as modernized in Cox [7], then explain Serre's strategy. -
Introduction to Class Field Theory and Primes of the Form X + Ny
Introduction to Class Field Theory and Primes of the Form x2 + ny2 Che Li October 3, 2018 Abstract This paper introduces the basic theorems of class field theory, based on an exposition of some fundamental ideas in algebraic number theory (prime decomposition of ideals, ramification theory, Hilbert class field, and generalized ideal class group), to answer the question of which primes can be expressed in the form x2 + ny2 for integers x and y, for a given n. Contents 1 Number Fields1 1.1 Prime Decomposition of Ideals..........................1 1.2 Basic Ramification Theory.............................3 2 Quadratic Fields6 3 Class Field Theory7 3.1 Hilbert Class Field.................................7 3.2 p = x2 + ny2 for infinitely n’s (1)........................8 3.3 Example: p = x2 + 5y2 .............................. 11 3.4 Orders in Imaginary Quadratic Fields...................... 13 3.5 Theorems of Class Field Theory.......................... 16 3.6 p = x2 + ny2 for infinitely many n’s (2)..................... 18 3.7 Example: p = x2 + 27y2 .............................. 20 1 Number Fields 1.1 Prime Decomposition of Ideals We will review some basic facts from algebraic number theory, including Dedekind Domain, unique factorization of ideals, and ramification theory. To begin, we define an algebraic number field (or, simply, a number field) to be a finite field extension K of Q. The set of algebraic integers in K form a ring OK , which we call the ring of integers, i.e., OK is the set of all α 2 K which are roots of a monic integer polynomial. In general, OK is not a UFD but a Dedekind domain. -
Units and Primes in Quadratic Fields
Units and Primes 1 / 20 Overview Evolution of Primality Norms, Units, and Primes Factorization as Products of Primes Units in a Quadratic Field 2 / 20 Rational Integer Primes Definition A rational integer m is prime if it is not 0 or ±1, and possesses no factors but ±1 and ±m. 3 / 20 Division Property of Rational Primes Theorem 1.3 Let p; a; b be rational integers. If p is prime and and p j ab, then p j a or p j b. 4 / 20 Gaussian Integer Primes Definition Let π; α; β be Gaussian integers. We say that prime if it is not 0, not a unit, and if in every factorization π = αβ, one of α or β is a unit. Note A Gaussian integer is a unit if there exists some Gaussian integer η such that η = 1. 5 / 20 Division Property of Gaussian Integer Primes Theorem 1.7 Let π; α; β be Gaussian integers. If π is prime and π j αβ, then π j α or π j β. 6 / 20 Algebraic Integers Definition An algebraic number is an algebraic integer if its minimal polynomial over Q has only rational integers as coefficients. Question How does the notion of primality extend to the algebraic integers? 7 / 20 Algebraic Integer Primes Let A denote the ring of all algebraic integers, let K = Q(θ) be an algebraic extension, and let R = A \ K. Given α; β 2 R, write α j β when there exists some γ 2 R with αγ = β. Definition Say that 2 R is a unit in K when there exists some η 2 R with η = 1. -
A Concrete Example of Prime Behavior in Quadratic Fields
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this paper is to provide a concise way for undergraduate math- ematics students to learn about how prime numbers behave in quadratic fields. This paper will provide students with some basic number theory background required to understand the material being presented. We start with the topic of quadratic fields, number fields of degree two. This section includes some basic properties of these fields and definitions which we will be using later on in the paper. The next section introduces the reader to prime numbers and how they are different from what is taught in earlier math courses, specifically the difference between an irreducible number and a prime number. We then move onto the majority of the discussion on prime numbers in quadratic fields and how they behave, specifically when a prime will ramify, split, or be inert. The final section of this paper will detail an explicit example of a quadratic field and what happens to prime numbers p within it. The specific field we choose is Q( −5) and we will be looking at what forms primes will have to be of for each of the three possible outcomes within the field. 2. Quadratic Fields One of the most important concepts of algebraic number theory comes from the factorization of primes in number fields. We want to construct Date: March 17, 2017. 1 2 CASEY BRUCK a way to observe the behavior of elements in a field extension, and while number fields in general may be a very complicated subject beyond the scope of this paper, we can fully analyze quadratic number fields. -
Computational Techniques in Quadratic Fields
Computational Techniques in Quadratic Fields by Michael John Jacobson, Jr. A thesis presented to the University of Manitoba in partial fulfilment of the requirements for the degree of Master of Science in Computer Science Winnipeg, Manitoba, Canada, 1995 c Michael John Jacobson, Jr. 1995 ii I hereby declare that I am the sole author of this thesis. I authorize the University of Manitoba to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Manitoba to reproduce this thesis by photocopy- ing or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. iii The University of Manitoba requires the signatures of all persons using or photocopy- ing this thesis. Please sign below, and give address and date. iv Abstract Since Kummer's work on Fermat's Last Theorem, algebraic number theory has been a subject of interest for many mathematicians. In particular, a great amount of effort has been expended on the simplest algebraic extensions of the rationals, quadratic fields. These are intimately linked to binary quadratic forms and have proven to be a good test- ing ground for algebraic number theorists because, although computing with ideals and field elements is relatively easy, there are still many unsolved and difficult problems re- maining. For example, it is not known whether there exist infinitely many real quadratic fields with class number one, and the best unconditional algorithm known for computing the class number has complexity O D1=2+ : In fact, the apparent difficulty of com- puting class numbers has given rise to cryptographic algorithms based on arithmetic in quadratic fields. -
Gaussian Prime Labeling of Super Subdivision of Star Graphs
of Math al em rn a u ti o c J s l A a Int. J. Math. And Appl., 8(4)(2020), 35{39 n n d o i i t t a s n A ISSN: 2347-1557 r e p t p n l I i c • Available Online: http://ijmaa.in/ a t 7 i o 5 n 5 • s 1 - 7 4 I 3 S 2 S : N International Journal of Mathematics And its Applications Gaussian Prime Labeling of Super Subdivision of Star Graphs T. J. Rajesh Kumar1,∗ and Antony Sanoj Jerome2 1 Department of Mathematics, T.K.M College of Engineering, Kollam, Kerala, India. 2 Research Scholar, University College, Thiruvananthapuram, Kerala, India. Abstract: Gaussian integers are the complex numbers of the form a + bi where a; b 2 Z and i2 = −1 and it is denoted by Z[i]. A Gaussian prime labeling on G is a bijection from the vertices of G to [ n], the set of the first n Gaussian integers in the spiral ordering such that if uv 2 E(G), then (u) and (v) are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of super subdivision of star graphs. MSC: 05C78. Keywords: Gaussian Integers, Gaussian Prime Labeling, Super Subdivision of Graphs. © JS Publication. 1. Introduction The graphs considered in this paper are finite and simple. The terms which are not defined here can be referred from Gallian [1] and West [2]. A labeling or valuation of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy, a label depending upon the vertex labels f(x) and f(y). -
Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai. -
On the Rational Approximations to the Powers of an Algebraic Number: Solution of Two Problems of Mahler and Mend S France
Acta Math., 193 (2004), 175 191 (~) 2004 by Institut Mittag-Leffier. All rights reserved On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mend s France by PIETRO CORVAJA and UMBERTO ZANNIER Universith di Udine Scuola Normale Superiore Udine, Italy Pisa, Italy 1. Introduction About fifty years ago Mahler [Ma] proved that if ~> 1 is rational but not an integer and if 0<l<l, then the fractional part of (~n is larger than l n except for a finite set of integers n depending on ~ and I. His proof used a p-adic version of Roth's theorem, as in previous work by Mahler and especially by Ridout. At the end of that paper Mahler pointed out that the conclusion does not hold if c~ is a suitable algebraic number, as e.g. 1 (1 + x/~ ) ; of course, a counterexample is provided by any Pisot number, i.e. a real algebraic integer c~>l all of whose conjugates different from cr have absolute value less than 1 (note that rational integers larger than 1 are Pisot numbers according to our definition). Mahler also added that "It would be of some interest to know which algebraic numbers have the same property as [the rationals in the theorem]". Now, it seems that even replacing Ridout's theorem with the modern versions of Roth's theorem, valid for several valuations and approximations in any given number field, the method of Mahler does not lead to a complete solution to his question. One of the objects of the present paper is to answer Mahler's question completely; our methods will involve a suitable version of the Schmidt subspace theorem, which may be considered as a multi-dimensional extension of the results mentioned by Roth, Mahler and Ridout. -
Number Theory Course Notes for MA 341, Spring 2018
Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads . -
An Euler Phi Function for the Eisenstein Integers and Some Applications
An Euler phi function for the Eisenstein integers and some applications Emily Gullerud aBa Mbirika University of Minnesota University of Wisconsin-Eau Claire [email protected] [email protected] January 8, 2020 Abstract The Euler phi function on a given integer n yields the number of positive integers less than n that are relatively prime to n. Equivalently, it gives the order of the group Z of units in the quotient ring (n) for a given integer n. We generalize the Euler phi function to the Eisenstein integer ring Z[ρ] where ρ is the primitive third root of 2πi/3 Z unity e by finding the order of the group of units in the ring [ρ] (θ) for any given Eisenstein integer θ. As one application we investigate a sufficiency criterion for Z × when certain unit groups [ρ] (γn) are cyclic where γ is prime in Z[ρ] and n ∈ N, thereby generalizing well-known results of similar applications in the integers and some lesser known results in the Gaussian integers. As another application, we prove that the celebrated Euler-Fermat theorem holds for the Eisenstein integers. Contents 1 Introduction 2 2 Preliminaries and definitions 3 2.1 Evenand“odd”Eisensteinintegers . 5 arXiv:1902.03483v2 [math.NT] 6 Jan 2020 2.2 ThethreetypesofEisensteinprimes . 8 2.3 Unique factorization in Z[ρ]........................... 10 3 Classes of Z[ρ] /(γn) for a prime γ in Z[ρ] 10 3.1 Equivalence classes of Z[ρ] /(γn)......................... 10 3.2 Criteria for when two classes in Z[ρ] /(γn) are equivalent . 13 3.3 Units in Z[ρ] /(γn)............................... -
THE GAUSSIAN INTEGERS Since the Work of Gauss, Number Theorists
THE GAUSSIAN INTEGERS KEITH CONRAD Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. The example we will look at in this handout is the Gaussian integers: Z[i] = fa + bi : a; b 2 Zg: Excluding the last two sections of the handout, the topics we will study are extensions of common properties of the integers. Here is what we will cover in each section: (1) the norm on Z[i] (2) divisibility in Z[i] (3) the division theorem in Z[i] (4) the Euclidean algorithm Z[i] (5) Bezout's theorem in Z[i] (6) unique factorization in Z[i] (7) modular arithmetic in Z[i] (8) applications of Z[i] to the arithmetic of Z (9) primes in Z[i] 1. The Norm In Z, size is measured by the absolute value. In Z[i], we use the norm. Definition 1.1. For α = a + bi 2 Z[i], its norm is the product N(α) = αα = (a + bi)(a − bi) = a2 + b2: For example, N(2 + 7i) = 22 + 72 = 53. For m 2 Z, N(m) = m2. In particular, N(1) = 1. Thinking about a + bi as a complex number, its norm is the square of its usual absolute value: p ja + bij = a2 + b2; N(a + bi) = a2 + b2 = ja + bij2: The reason we prefer to deal with norms on Z[i] instead of absolute values on Z[i] is that norms are integers (rather than square roots), and the divisibility properties of norms in Z will provide important information about divisibility properties in Z[i]. -
Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases
INTERSECTIONS OF DELETED DIGITS CANTOR SETS WITH GAUSSIAN INTEGER BASES Athesissubmittedinpartialfulfillment of the requirements for the degree of Master of Science By VINCENT T. SHAW B.S., Wright State University, 2017 2020 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES May 1, 2020 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPER- VISION BY Vincent T. Shaw ENTITLED Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases BE ACCEPTED IN PARTIAL FULFILLMENT OF THE RE- QUIREMENTS FOR THE DEGREE OF Master of Science. ____________________ Steen Pedersen, Ph.D. Thesis Director ____________________ Ayse Sahin, Ph.D. Department Chair Committee on Final Examination ____________________ Steen Pedersen, Ph.D. ____________________ Qingbo Huang, Ph.D. ____________________ Anthony Evans, Ph.D. ____________________ Barry Milligan, Ph.D. Interim Dean, School of Graduate Studies Abstract Shaw, Vincent T. M.S., Department of Mathematics and Statistics, Wright State University, 2020. Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases. In this paper, the intersections of deleted digits Cantor sets and their fractal dimensions were analyzed. Previously, it had been shown that for any dimension between 0 and the dimension of the given deleted digits Cantor set of the real number line, a translate of the set could be constructed such that the intersection of the set with the translate would have this dimension. Here, we consider deleted digits Cantor sets of the complex plane with Gaussian integer bases and show that the result still holds. iii Contents 1 Introduction 1 1.1 WhystudyintersectionsofCantorsets? . 1 1.2 Prior work on this problem . 1 2 Negative Base Representations 5 2.1 IntegerRepresentations .............................