DOCSLIB.ORG

A Computational Exploration of Gaussian and Eisenstein Moats

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

A Computational Exploration of Gaussian and Eisenstein Moats By Philip P. West A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS IN SCIENCE (MATHEMATICS) at the CALIFORNIA STATE UNIVERSITY CHANNEL ISLANDS 20 14 copyright 20 14 Philip P. West ALL RIGHTS RESERVED APPROVED FOR THE MATHEMATICS PROGRAM Brian Sittinger, Advisor Date: 21 December 20 14 Ivona Grzegorzyck Date: 12/2/14 Ronald Rieger Date: 12/8/14 APPROVED FOR THE UNIVERSITY Doctor Gary A. Berg Date: 12/10/14 Non- Exclusive Distribution License In order for California State University Channel Islands (C S U C I) to reproduce, translate and distribute your submission worldwide through the C S U C I Institutional Repository, your agreement to the following terms is necessary. The authors retain any copyright currently on the item as well as the ability to submit the item to publishers or other repositories. By signing and submitting this license, you (the authors or copyright owner) grants to C S U C I the nonexclusive right to reproduce, translate (as defined below), and/ or distribute your submission (including the abstract) worldwide in print and electronic format and in any medium, including but not limited to audio or video. You agree that C S U C I may, without changing the content, translate the submission to any medium or format for the purpose of preservation. You also agree that C S U C I may keep more than one copy of this submission for purposes of security, backup and preservation. You represent that the submission is your original work, and that you have the right to grant the rights contained in this license. You also represent that your submission does not, to the best of your knowledge, infringe upon anyone's copyright. You also represent and warrant that the submission contains no libelous or other unlawful matter and makes no improper invasion of the privacy of any other person. If the submission contains material for which you do not hold copyright, you represent that you have obtained the unrestricted permission of the copyright owner to grant C S U C I the rights required by this license, and that such third party owned material is clearly identified and acknowledged within the text or content of the submission. You take full responsibility to obtain permission to use any material that is not your own. This permission must be granted to you before you sign this form. IF THE SUBMISSION IS BASED UPON WORK THAT HAS BEEN SPONSORED OR SUPPORTED BY AN AGENCY OR ORGANIZATION OTHER THAN C S U C I, YOU REPRESENT THAT YOU HAVE FULFILLED ANY RIGHT OF REVIEW OR OTHER OBLIGATIONS REQUIRED BY SUCH CONTRACT OR AGREEMENT. The C S U C I Institutional Repository will clearly identify your names as the authors or owners of the submission, and will not make any alteration, other than as allowed by this license, to your submission. Title of Item: A comutational exploration of Gaussian and Eisenstein Moats 3 to 5 keywords or phrases to describe the item: Gaussian Moat, Eisenstein Moat, Eisenstein Prime Density Authors Name (Print): Philip P. West Authors Signature Date: Dec 5, 20 14 This is a permitted, modified version of the Non- exclusive Distribution License from M I T Libraries and the University of Kansas. DEDICATION I dedicate this paper to the following two influential women in my life. First, I acknowledge my wife Sharon Hines- West who has always been supportive of me, both with her encouragement and with her own knowledge of mathemat- ics. Secondly, I am grateful to my mother Doctor Julia S. West, who despite many personal struggles, received her doctorate at age 58 and taught her entire life right up to her passing at age 81. Her example inspired me to pursue this goal later in life. roman numeral 4 ACKNOWLEDGEMENTS This thesis would not have been possible without the support of the Mathe- matics Department at California State University Channel Islands. In partic- ular, I would like to thank my advisors Doctor Brian Sittinger, Ronald Rieger, and Doctor. Ivona Grzegorczyk. I also would like to acknowledge Doctor Jorge Garcia for his help formatting this thesis using Latex. Camarillo, California December 3, 20 14 v ABSTRACT If one imagines the Gaussian primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture. If the frog's journey terminates for a given hop size, it implies that a prime free "moat" greater than the hop size completely surrounds the origin. In the Chauvenet Prize- winning paper "A Stroll Through the Gaussian Primes", Ellen Gethner, Stan Wagon, and Brian Wick [4] explored this problem and by computational methods proved the existence of a square root of 26 -moat. Additionally they proved that prime-free neighborhoods of arbitrary radius k surrounding a Gaussian prime exist. In their concluding remarks, Gethner et al. note that "Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1." 1 This paper takes up that challenge by examining similar questions about walks to infinity in the ring of Eisenstein integers. By computational methods, we prove the existence of a bounding square root of l6 -moat in the Eisenstein primes and provide an estimate for the bounds of any square root of k -moat. A subtle difference be- tween the structure of Eisenstein and Gaussian integers results in our estimate being significantly different from our initial expectation. 2 TABLE OF CONTENTS 1 Introduction 5 2 Mathematical Background 9 2 point 1 Arithmetic in the Gaussian and Eisenstein Integers 9 2 point 2 The Chinese Remainder Theorem 12 2 point 3 Gaussian Moats (Gethner) 13 2 point 4 Gaussian Prime Density (Vardi) 17 2 point 5 Estimating Bounds on Gaussian Moats (Vardi) 18 3 Computational Background 21 3 point 1 Improved Computational Results (Tsuchimura) 21 3 point 2 Depth-first Search Algorithm 25 3 point 3 Multiple Precision Integers and Rationals Library 26 3 point 4 Miller- Rabin Primality Testing 27 4 Mathematical Results 31 4 point 1 Annular Moats 31 4 point 2 Eisenstein Prime Density 33 3 TABLE OF CONTENTS 4 point 3 Finding p parenthesis k parenthesis for a k- connected Eisenstein Prime Component 35 4 point 4 Estimating Methods for Bounding Eisenstein Moats 37 4 point 5 Generalizing Gethner's Proof to the Eisenstein Integers 39 5 Computational Results 45 5 point 1 Overview of Approach 45 5 point 2 k- Connected Gaussian Prime Algorithm 46 5 point 3 k- Connected Eisenstein Prime Algorithm 50 5 point 4 Gaussian Prime Viewer for Validation Purposes 54 5 point 5 Validation of Gaussian Moat Data 56 5 point 6 Eisenstein Moat Data 57 6 Conclusion 60 7 Appendix 65 7 point 1 Gethner's Proof Generalized for Imaginary Quadratic Integers 65 7 point 2 MAPLE Code for square root of 7-connected E- Prime Plot 71 7 point 3 C- code for Gaussian Prime Viewer 72 7 point 4 C- code for Eisenstein Prime Connected Components 77 Bibliography 87 4 Chapter 1 INTRODUCTION Simple problems often lead to more interesting and difficult questions. For instance, is it possible to travel on the real number line to infinity by taking bounded steps on only the prime numbers? The answer is no and easily seen by showing arbitrarily large prime-free gaps exist on the number line by ob- serving that for any integer n greater than 1, parenthesis n plus 1 parenthesis exclamation mark plus 2, parenthesis n plus 1 parenthesis exclamation mark plus 3, dot dot dot, parenthesis n plus 1 parenthesis exclamation mark plus parenthesis n plus 1 parenthesis are n consecutive composite integers. Now that we know arbitrarily large gaps without primes exist on the real number line, then inquiring minds would ask, "Are there also Gaussian prime-free moats of arbitrarily large width that surround the origin in the complex plane?" The answer is, "Probably, but we have not been able to prove it." 5 A problem more analogous to the question about arbitrary prime free- gaps on the real line is whether two- dimensional prime-free neighborhoods of arbitrary radius k exist. This was proven by Gethner et al. in 19 98, [4]. We provide a direct modification of this proof from the Gaussian integers to the Eisenstein integers. Definition 1 point 0 point 1 (k- connected component) A set of vertices in a graph whose vertices to each other by edges of length k or less. We will only consider the set of vertices representing Gaussian primes or the set of vertices representing Eisen- stein primes in the complex plane. In both cases, our connected component will always start with the element closest to the origin of the complex plane. Gethner et al. provided computational proofs for the existence of Gaussian moats with the specific maximum distance reachable for all connected com- ponents surrounded by moats up to square root of 20. Additionally, they computationally demonstrated that a square root of 26 -moat exists, but only provided a bound for the max- imum reachable distance of its connected component. Sometimes, applied mathematics can provide insights for solving abstract problems. For instance, Percolation Theory, which describes the behavior of connected clusters in a random graph, is used in material science to model 6 movement and filtering of fluids through porous materials.
Recommended publications
  • Declaration of Independence Font Style

    Declaration of Independence Font Style

    Declaration Of Independence Font Style Heliocentric and implanted Jeremiah prize her duteousness cannonades while Tabor numerate some Davy tonally. How integrant is Juergen when natatory and denticulate Xenos balk some inmate? Outsize and unmeant Shimon spottings fundamentally and synthesise his loungers ghastfully and aguishly. Headings should be closer to the text they introduce than the text that preceeds them. Son foundry was divided among his heirs. HTML hyperlinks are automatically converted. Need help finding the right font for your brand? Prince currently defaults to the RGB color space. Characters are in this declaration independence calligraphy font was the lanston caslon. Mulberry comes as a group of six fonts that cover a whole array of different styles and ligatures. By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. Now check your email to confirm your subscription. Research on font trustworthiness: Baskerville vs. Files included in attentions to hear from the depository of? Writers used both cursive styles: location, contents and context of the text determined which style to use. In general, although some of the shapes of individual characters are different, the biggest variation is in line spacing and character size. Some fonts give off assertiveness. The Declaration of Independence in its popular calligraphic form, with signatures. Should this be of importance, use the second approach instead. There are several bibles that are set with Lexicon as well. Garamond will undoubtedly fit the bill. However, it will be tricky to support nested styles this way. Who are your people, and how do you want to talk to them? QUILTSportraits of famous African Americans as a way to document and commemorate their achievements.
  • Gaussian Prime Labeling of Super Subdivision of Star Graphs

    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).
  • 19. Punctuation

    19. Punctuation

    punctuation 19. Punctuation Punctuation is important because it helps achieve clarity and readability . ’ Apostrophes Use • before the “s” in singular possessives: The prime minister’s suggestion was considered. • after the “s” in plural possessives: The ministers’ decision was unanimous. Do not use • in plural dates and abbreviations: 1930s, NGOs • in the possessive pronoun “its”: The government characterised its budget as prudent. See also: Capitalisation, pp. 66-68. : Colons Use • to lead into a list, an explanation or elaboration, an indented quotation • to mark the break between a title and subtitle: Social Sciences for a Digital World: Building Infrastructure for the Future (book) Trends in transport to 2050: A macroscopic view (chapter) Do not use • more than once in a given sentence • a space before colons and semicolons. 90 oecd style guide - third edition @oecd 2015 punctuation , Commas Use • to separate items in most lists (except as indicated under semicolons) • to set off a non-restrictive relative clause or other element that is not part of the main sentence: Mr Smith, the first chairperson of the committee, recommended a fully independent watchdog. • commas in pairs; be sure not to forget the second one • before a conjunction introducing an independent clause: It is one thing to know a gene’s chemical structure, but it is quite another to understand its actual function. • between adjectives if each modifies the noun alone and if you could insert the word “and”: The committee recommended swift, extensive changes. Do not use • after “i.e.” or “e.g.” • before parentheses • preceding and following en-dashes • before “and”, at the end of a sequence of items, unless one of the items includes another “and”: The doctor suggested an aspirin, half a grapefruit and a cup of broth.
  • The Group of Classes of Congruent Quadratic Integers with Respect to Any Composite Ideal Modulus Whatever

    The Group of Classes of Congruent Quadratic Integers with Respect to Any Composite Ideal Modulus Whatever

    THE GROUPOF CLASSESOF CONGRUENTQUADRATIC INTEGERS WITH RESPECTTO A COMPOSITEIDEAL MODULUS* BY ARTHUR RANUM Introduction. If in the ordinary theory of rational numbers we consider a composite integer m as modulus, and if from among the classes of congruent integers with respect to that modulus we select those which are prime to the modulus, they form a well-known multiplicative group, which has been called by Weber (Algebra, vol. 2, 2d edition, p. 60), the most important example of a finite abelian group. In the more general theory of numbers in an algebraic field we may in a corre- sponding manner take as modulus a composite ideal, which includes as a special case a composite principal ideal, that is, an integer in the field, and if we regard all those integers of the field which are congruent to one another with respect to the modulus as forming a class, and if we select those classes whose integers are prime to the modulus, they also will form a finite abelian group f under multiplication. The investigation of the nature of this group is the object of the present paper. I shall confine my attention, however, to a quadratic number-field, and shall determine the structure of the group of classes of congruent quadratic integers with respect to any composite ideal modulus whatever. Several distinct cases arise depending on the nature of the prime ideal factors of the modulus ; for every case I shall find a complete system of independent generators of the group. Exactly as in the simpler theory of rational numbers it will appear that the solution of the problem depends essentially on the case in which the modulus is a prime-power ideal, that is, a power of a prime ideal.
  • Number Theory Course Notes for MA 341, Spring 2018

    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

    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 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 .............................
  • UEB Guidelines for Technical Material

    UEB Guidelines for Technical Material

    Guidelines for Technical Material Unified English Braille Guidelines for Technical Material This version updated October 2008 ii Last updated October 2008 iii About this Document This document has been produced by the Maths Focus Group, a subgroup of the UEB Rules Committee within the International Council on English Braille (ICEB). At the ICEB General Assembly in April 2008 it was agreed that the document should be released for use internationally, and that feedback should be gathered with a view to a producing a new edition prior to the 2012 General Assembly. The purpose of this document is to give transcribers enough information and examples to produce Maths, Science and Computer notation in Unified English Braille. This document is available in the following file formats: pdf, doc or brf. These files can be sourced through the ICEB representatives on your local Braille Authorities. Please send feedback on this document to ICEB, again through the Braille Authority in your own country. Last updated October 2008 iv Guidelines for Technical Material 1 General Principles..............................................................................................1 1.1 Spacing .......................................................................................................1 1.2 Underlying rules for numbers and letters.....................................................2 1.3 Print Symbols ..............................................................................................3 1.4 Format.........................................................................................................3
  • Gaussian Integers

    Gaussian Integers

    Gaussian integers 1 Units in Z[i] An element x = a + bi 2 Z[i]; a; b 2 Z is a unit if there exists y = c + di 2 Z[i] such that xy = 1: This implies 1 = jxj2jyj2 = (a2 + b2)(c2 + d2) But a2; b2; c2; d2 are non-negative integers, so we must have 1 = a2 + b2 = c2 + d2: This can happen only if a2 = 1 and b2 = 0 or a2 = 0 and b2 = 1. In the first case we obtain a = ±1; b = 0; thus x = ±1: In the second case, we have a = 0; b = ±1; this yields x = ±i: Since all these four elements are indeed invertible we have proved that U(Z[i]) = {±1; ±ig: 2 Primes in Z[i] An element x 2 Z[i] is prime if it generates a prime ideal, or equivalently, if whenever we can write it as a product x = yz of elements y; z 2 Z[i]; one of them has to be a unit, i.e. y 2 U(Z[i]) or z 2 U(Z[i]): 2.1 Rational primes p in Z[i] If we want to identify which elements of Z[i] are prime, it is natural to start looking at primes p 2 Z and ask if they remain prime when we view them as elements of Z[i]: If p = xy with x = a + bi; y = c + di 2 Z[i] then p2 = jxj2jyj2 = (a2 + b2)(c2 + d2): Like before, a2 + b2 and c2 + d2 are non-negative integers. Since p is prime, the integers that divide p2 are 1; p; p2: Thus there are three possibilities for jxj2 and jyj2 : 1.
  • Algebraic Number Theory Part Ii (Solutions) 1

    Algebraic Number Theory Part Ii (Solutions) 1

    ALGEBRAIC NUMBER THEORY PART II (SOLUTIONS) 1. Eisenstein Integers Exercise 1. Let p −1 + −3 ! = : 2 Verify that !2 + ! + 1 = 0. Solution. We have p 2 p p p −1 + −3 −1 + −3 1 + −3 −1 + −3 + + 1 = − + + 1 2 2 2 2 1 1 1 1 p = − − + 1 + − + −3 2 2 2 2 = 0: Exercise 2. The set Z[!] = fa + b! : a; b 2 Zg is called the ring of Eisenstein integers. Prove that it is a ring. That is, show that for any Eisenstein integers a + b!, c + d! the numbers (a + b!) + (c + d!); (a + b!) − (c + d!); (a + b!)(c + d!) are Eisenstein integers as well. Solution. For the sum and difference the calculations are easy: (a + b!) + (c + d!) = (a + c) + (b + d)!; (a + b!) − (c + d!) = (a − c) + (b − d)!: Since a; b; c; d are integers, we see that a + c; b + d; a − c; b − d are integers as well. Therefore the above numbers are Eisenstein integers. The product is a bit more tricky. Here we need to use the fact that !2 = −1−!: (a + b!)(c + d!) = ac + ad! + bc! + bd!2 = ac + ad! + bc! + bd(−1 − !) = (ac − bd) + (ad + bc − bd)!: Since a; b; c; d are integers, we see that ac − bd and ad + bc − bd are integers as well, so the above number is an Einstein integer. Exercise 3. For any Eisenstein integer a + b! define the norm N(a + b!) = a2 − ab + b2: Prove that the norm is multiplicative. That is, for any Eisenstein integers a + b!, c + d! the equality N ((a + b!)(c + d!)) = N(a + b!)N(c + d!) 1 2 ALGEBRAIC NUMBER THEORY PART II (SOLUTIONS) holds.
  • Not-So-Frequently Asked Questions for LATEX

    Not-So-Frequently Asked Questions for LATEX

    Not-So-Frequently Asked Questions for LATEX Miles 2010 This document addresses more esoteric issues in LATEX that have nonethe- less actually arisen with the author. We hope that somebody will find it useful! • Why is LATEX telling me that a command I use is undefined? Answer. Make sure that you're using the amsmath package. This is included in rsipacks.sty (so it will automatically be included in your RSI paper and minipaper). To include it in other documents, put \usepackage{amsmath} in your preamble, before \begin{document}. Just to be safe, throw in amssymb and amsthm as well (so that you have all the fonts and symbols you would expect, and so that you can define theorems environments). If the command is a standard one, be sure that you spelled it correctly. If it's a command you defined (or thought you did), make sure that you really defined it. • How do I get script letters, like L , H , F , and G ? Answer. You need the mathrsfs package, so put \usepackage{mathrsfs} in your preamble. Then you can use the font name mathscr to get the desired font in math mode. For example, \mathscr{L} yields L . • How do I typeset a series of displayed equations so that the equals signs line up? Answer. First, you need the amsmath package (see above). Once you have that, you use the align* environment, like this : \begin{align*} math &= more math \\ &= more math \\ 1 other math &\le different math \\ &= yet more math \end{align*} This will produce something like n i X X X f(i; j) = f(i; j) i=1 j=1 1≤j≤i≤n n n X X = f(i; j); j=1 i=j 1 + 1 + 1 = 2 + 1 = 3: You can replace the equals signs with whatever other appropriate sym- bol you like (≤, ≥, ≡, =∼, ⊂, etc.).