DOCSLIB.ORG

INSTITUTE of TECHNOLOGY SLIGO Coding Theory, Group

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
INSTITUTE of TECHNOLOGY SLIGO Coding Theory, Group INSTITUTE OF TECHNOLOGY SLIGO Coding Theory, Group Automorphisms and Units of Finite Group Algebras Fergal Gallagher This project is submitted in fulfilment of the requirements for the Degree of Master of Engineering Supervisor: Dr. Leo Creedon Submitted to the Institute of Technology, Sligo July 2014 Declaration I certify that the material which I submit for assessment leading to the award of MEng is entirely my own work and has not been taken from others, save the extent that such work has been cited and acknowledged within the text of the study. .................................................. Fergal Gallagher 2 Contents Abstract ii Acknowledgements iv Frequently Used Notation v 1 Introduction 1 1.1 Cyclic Codes . 7 1.2 Group Ring Matrices . 9 1.3 Zero Divisor Codes . 11 2 Automorphisms of Finite Abelian Groups 14 2.1 Automorphisms of Cyclic Groups Cp and Cpn . 17 2.2 Automorphisms of Elementary Abelian groups . 22 n 2.3 Automorphisms of Cpα ...................... 24 2.4 Automorphisms of Cpm × Cpn . 25 2.5 Automorphisms of abelian p − groups with repeating factors . 29 2.6 Finding jAut(Hp)j using Endomorphism Rings . 31 2.7 Table of Aut(G) for Small Abelian Groups . 40 3 Automorphisms of Non-Abelian Groups 44 3.1 Semidirect products . 49 3.2 Automorphisms of Dihedral Groups . 52 3.3 Automorphisms of General Linear Groups . 55 3.4 Automorphism Groups of Automorphism Groups . 59 3.5 Automorphism Tower for Small Groups . 61 4 Finite Commutative Group Algebras 62 4.1 The Unit Group of FG ...................... 63 5 U(FG) where F has char p and G is an abelian p − group 83 5.1 U(FG) where char(F ) = 2 and G is an abelian 2 − group . 86 i 5.2 U(FG) where char(F ) = p and G is an abelian p − group . 89 6 Idempotents and the Decomposition of Semisimple Group Algebras 102 6.1 Field Automorphisms . 106 6.2 Group Representations and Group Characters . 110 6.3 Cyclotomic Classes of G and the Primitive Central Orthogonal Idempotents of FG ........................112 6.4 The Isomorphism Problem . 131 6.5 Table of U(FG) for selected group algebras (G abelian) . 140 7 Perlis Walker Theorem - Finding U(FG) 142 7.1 A General Approach . 145 7.2 U(FG) where Maschke's Theorem does not apply . 149 7.3 The Unit Groups . 150 7.4 General Table of U(FG)(G abelian) . 167 8 Conclusion 176 References 179 Bibliography 182 ii Abstract Group Algebras can be used to construct Low Density Parity Check Codes (LDPC) and Convolution Codes. These codes have applications within digi- tal communication and storage, such as to improve the performance of digital radio, digital video, mobile phones, satellite and deep space communications, as well as bluetooth implementations. In this Masters Thesis, theoretical mathematical techniques are used to construct an atlas of finite group algebras. In particular, we find and list the automorphism group of abelian groups and the unit group of finite commu- tative group algebras. The aim of this atlas is to improve our understanding of group algebras and their applications to Coding Theory. Firstly, the basic concepts of coding theory are introduced. The next section of this thesis (Chapter 2- Automorphisms of Finite Abelian Groups) deals with various techniques for finding the automorphism group of different categories of abelian groups. In particular, where the group is an abelian p − group with 2 distinct direct factors, use is made of recent techniques by Bidwell and Curran (2010). Hillar and Rhea (2007) give a technique involving endomorphism rings which allows the calculation of the order of Aut(G) where G is abelian. Using these techniques and others a table is presented giving the structure and order of the automorphism group for many abelian groups. Chapter 3 (Automorphisms of Non-Abelian Groups) looks beyond abelian groups. Recent methods by Curran (2008) using crossed homomorphisms are used where G is a semidirect product. Dihedral groups and general linear groups are also examined. At the end of this section there are some conjectures relating to the automorphisms of groups in general and a table is presented showing the automorphism tower of small groups. The next section of the thesis (Chapter 4 - Finite Commutative Group Algebras) introduces the concepts of group algebras and unit groups. This Chapter contains many specific example of group algebras. In these examples the structure and order of the unit groups are examined. iii In Chapter 5 (U(FG) where F has char p and G is a p − group), a technique is presented for finding the structure of the unit group for non- Maschke cases. This technique involves counting the number of elements in the normalised unit group which have order dividing a particular power of p for group algebras of the form FG where G is a p − group and F is a field of characteristic p. This Chapter concludes with a Theorem which gives the unit group of all group algebras of the above form. There are also some examples illustrating this. Chapter 6 (Idempotents and the decomposition of FG) then looks at ways of finding the Artin Wedderburn decomposition where applicable. Here, re- cent techniques by Broche and Del Rio (2007) are used to find the decompo- sition and also to find the primitive central idempotents. Finally, in Chapter 7, The Perlis Walker Theorem (1950) is used and adapted to give more general results for all possible group algebras for abelian groups. This leads to a general table giving the decomposition and unit groups of the group algebras for all abelian groups of order up to 15. In doing this, we get a further insight into the isomorphism problem for group algebras. This includes the result that given two non-isomorphic abelian groups G and H each with order n, and a field F of order q such that q ≡ 1(mod n), then FG ' FH. Thus there is a whole class of isomorphic group algebras of this type and in each of these instances the decomposition is the direct product of n copies of the field F . We show that the minimal isomorphic pair of group algebras FG and FH with G and H not isomorphic which is not of this type is F5C12 and F5(C2 × C6). We also show that there is yet another class of isomorphic group algebras. Given two non-isomorphic abelian groups G and H each with order n and each containing m elements of order 2, and a field F of order q such that q ≡ -1(mod e) where e is the Lm exponent of the group, then FG ' FH. In this case, FG ' FH ' i=1 Fq L[n−(m)]=2 3 ⊕ i=1 Fq2 . An example of this is F7(C2 × C4 × C8) ' F7(C4 ). iv Acknowledgements First of all, thanks to my wonderful wife Shona Heffernan for supporting me while I studied. She must often have wondered when or if I would ever finish. Also to my parents, Joan and Lionel, and to Margaret and Padraic Heffernan I am very grateful for helping out in so many ways. Second of all thanks to Grace Corcoran, who met me for coffee many moons ago and put me in touch with her colleague Leo Creedon who became my supervisor for this project. Thanks to Leo for introducing me to linear and abstract algebra, a subject I knew nothing about and a subject that has fascinated me ever since. Thanks for listening to my questions and answering usually with another question. It has stood me in good stead. Apologies for scribbling in the margins of your T opics in Algebra book Leo! Thanks to Ian McLoughlin and Faye Monaghan for their support and encouragement and for adding a new vibrancy to IT Sligo's mathematical research. To my colleages in the research lab (some of whom have escaped already), thanks for helping me to do battle with printers, binders, computers and sandwichmakers and for providing lots of laughs along the way. Thanks also to Aoife, Carmel, Catriona and Taraneh in the Engineering Office who are always happy to assist. Finally thanks to John Bartlett and Mary McLoughlin in the research office for all of their work and to Sligo County Council for contributing to my college fees. v Frequently Used Notation C a code c a codeword G a generator matrix H a parity check matrix w(c) the minimum weight of c dmin the minimum distance of a code n (n; k) code a k dimensional subspace of Fq (n; k; d) code an (n; k) code with minimum distance d p a prime integer n a positive integer Z the set of integers N the set of positive integers a(mod n) the remainder when a is divided by n (a; b) the greatest common divisor of a; b φ(n) Euler's totient function - the number of a ≤ n such that (a; n) = 1 G a group jGj the order of the group G hxi the group generated by x xy yxy−1, the conjugate of x by y jxj o(x), the order of the group element x H < G H is a subgroup of G H/GH is a normal subgroup of G G=H the group of cosets of the normal subgroup H G × H the direct product of groups G and H N o H the semidirect product of groups N and H with N a normal subgroup Cn the cyclic group of order n k Cn the direct product of k copies of the cyclic group of order n vi 1 ≤ i ≤ n Dn the dihedral group of order n Sn the symmetric group on n objects Z(G) the centre of G G0 the commutator subgroup of G Inn(G) the group of inner automorphisms of G Aut(G) the group of automorphisms of G Aut(G : H) the group of automorphisms of G which fix the subgroup H Aut0GG AutnG Aut(Autn−1G) χ an irreducible character of a group G∗ the group of irreducible characters of G R a ring F a field Fn the finite field with n elements Z=nZ the ring of integers mod n (Z=nZ)× the multiplicative group of units of the ring Z=nZ F [x] the algebra of polynomials in x with coefficients in F GLn(F ) the general linear group of degree n over the field F GLn(Z=mZ) the general linear group of degree n over Z=mZ
Load more

Recommended publications
  • Complete Objects in Categories
    Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion.
    [Show full text]
  • MATH 412 PROBLEM SET 11, SOLUTIONS Reading: • Hungerford
    MATH 412 PROBLEM SET 11, SOLUTIONS Reading: • Hungerford 8.4, 9.1, 9.2 Practice Problems: These are “easier” problems critical for your understanding, but not to be turned in. Be sure you can do these! You may see them or something very similar on Quizzes and Exams. Section 8.4: 9, 19, 21, Section 9.1: 1, 2, 3, 4, 5 Section 9.2: 1, 3 Problems due Friday. Write out carefully, staple solutions to A, B and C in one pack, and stable solutions to D and E in another. Write your name and section number on each. (A) Let F be a field. Consider the group SL2(F) of 2 × 2 matrices with entries in F and determinant one. 1 (1) Show that the center of SL2(F) is Z = {±Ig, where I is the 2 × 2 identity matrix. (2) Let PSL2(F) be the quotient group SL2(F)=Z. Compute the order of PSL2(Z7) as a group. 2 (3) For each of the following, compute its order as an element of PSL2(Z7). 0 2 • A = ± 3 0 1 1 • B = ± 6 0 1 3 • C = ± 0 1 Note: PSL2(Z7) is the smallest simple group that is neither cyclic nor isomorphic to An for some n. Solution. a b 1 1 1 0 (1) Let A = be an arbitrary element of SL ( ), B = , and C = . We c d 2 F 0 1 1 1 −c a − d b 0 compute AB − BA = and AC − CA = . If A 2 Z, then AB − BA = 0 c d − a −b AC − CA = 0, so b = c = a − d = 0.
    [Show full text]
  • The Number 6
    The number 6 Gabriel Coutinho 2013 Abstract The number 6 has a unique and distinguished property. It is the only natural number n for which there is a construction of n isomor- phic objects on a set with n elements, invariant under all permutations of the set, but not in natural bijection with the set. I will talk about this fact and a closely related result. This is based on Chapter 6 of Cameron and Van Lint [1]. 1 A distinguished property of 6 Theorem 1. The number 6 is the only natural number n that satisfies the following property: If E is an n-set, then there is a construction of exactly n isomor- phic objects on E whose union is invariant under the permuta- tions of E, but none correspond in natural way to an element of E. Some clarification is needed here. Neither the meaning of object nor that of natural is clear. I will make it even worse, and provide a bad definition for one, and an incomplete for the other. An object on a set E is a construction that can be made using the elements of E and set-theoretical operations, such as taking the union. An object on E corresponds naturally to a point of E if the automorphism group of the object is the stabilizer of that point in Sn. Despite such abstract formulation, we will see that this theorem has a very concrete interpretation. In particular, the \shape" of such objects will arise naturally throughout the proof. The idea of the proof is to analyse the automorphism group the objects defined by 1 the automorphisms of the ground set which do not change the object.
    [Show full text]
  • Elementary Number Theory
    Elementary Number Theory Peter Hackman HHH Productions November 5, 2007 ii c P Hackman, 2007. Contents Preface ix A Divisibility, Unique Factorization 1 A.I The gcd and B´ezout . 1 A.II Two Divisibility Theorems . 6 A.III Unique Factorization . 8 A.IV Residue Classes, Congruences . 11 A.V Order, Little Fermat, Euler . 20 A.VI A Brief Account of RSA . 32 B Congruences. The CRT. 35 B.I The Chinese Remainder Theorem . 35 B.II Euler’s Phi Function Revisited . 42 * B.III General CRT . 46 B.IV Application to Algebraic Congruences . 51 B.V Linear Congruences . 52 B.VI Congruences Modulo a Prime . 54 B.VII Modulo a Prime Power . 58 C Primitive Roots 67 iii iv CONTENTS C.I False Cases Excluded . 67 C.II Primitive Roots Modulo a Prime . 70 C.III Binomial Congruences . 73 C.IV Prime Powers . 78 C.V The Carmichael Exponent . 85 * C.VI Pseudorandom Sequences . 89 C.VII Discrete Logarithms . 91 * C.VIII Computing Discrete Logarithms . 92 D Quadratic Reciprocity 103 D.I The Legendre Symbol . 103 D.II The Jacobi Symbol . 114 D.III A Cryptographic Application . 119 D.IV Gauß’ Lemma . 119 D.V The “Rectangle Proof” . 123 D.VI Gerstenhaber’s Proof . 125 * D.VII Zolotareff’s Proof . 127 E Some Diophantine Problems 139 E.I Primes as Sums of Squares . 139 E.II Composite Numbers . 146 E.III Another Diophantine Problem . 152 E.IV Modular Square Roots . 156 E.V Applications . 161 F Multiplicative Functions 163 F.I Definitions and Examples . 163 CONTENTS v F.II The Dirichlet Product .
    [Show full text]
  • Arxiv:Math/0412262V2 [Math.NT] 8 Aug 2012 Etrgae Tgte Ihm)O Atnscnetr and Conjecture fie ‘Artin’S Number on of Cojoc Me) Domains’
    ARTIN’S PRIMITIVE ROOT CONJECTURE - a survey - PIETER MOREE (with contributions by A.C. Cojocaru, W. Gajda and H. Graves) To the memory of John L. Selfridge (1927-2010) Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the lit- erature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contribu- tions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Woj- ciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean domains’. Contents 1. Introduction 2 2. Naive heuristic approach 5 3. Algebraic number theory 5 3.1. Analytic algebraic number theory 6 4. Artin’s heuristic approach 8 5. Modified heuristic approach (`ala Artin) 9 6. Hooley’s work 10 6.1. Unconditional results 12 7. Probabilistic model 13 8. The indicator function 17 arXiv:math/0412262v2 [math.NT] 8 Aug 2012 8.1. The indicator function and probabilistic models 17 8.2. The indicator function in the function field setting 18 9. Some variations of Artin’s problem 20 9.1. Elliptic Artin (by A.C. Cojocaru) 20 9.2. Even order 22 9.3. Order in a prescribed arithmetic progression 24 9.4. Divisors of second order recurrences 25 9.5. Lenstra’s work 29 9.6.
    [Show full text]
  • Automorphisms of Group Extensions
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 155, Number t, March 1971 AUTOMORPHISMS OF GROUP EXTENSIONS BY CHARLES WELLS Abstract. If 1 ->■ G -í> E^-Tt -> 1 is a group extension, with i an inclusion, any automorphism <j>of E which takes G onto itself induces automorphisms t on G and a on n. However, for a pair (a, t) of automorphism of n and G, there may not be an automorphism of E inducing the pair. Let à: n —*■Out G be the homomorphism induced by the given extension. A pair (a, t) e Aut n x Aut G is called compatible if a fixes ker á, and the automorphism induced by a on Hü is the same as that induced by the inner automorphism of Out G determined by t. Let C< Aut IT x Aut G be the group of compatible pairs. Let Aut (E; G) denote the group of automorphisms of E fixing G. The main result of this paper is the construction of an exact sequence 1 -» Z&T1,ZG) -* Aut (E; G)-+C^ H*(l~l,ZG). The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless "exact" at C in the obvious sense. 1. Notation. If G is a group with subgroup H, we write H<G; if H is normal in G, H<¡G. CGH and NGH are the centralizer and normalizer of H in G. Aut G, Inn G, Out G, and ZG are the automorphism group, the inner automorphism group, the outer automorphism group, and the center of G, respectively.
    [Show full text]
  • On Inner Automorphisms of Finite Groups
    proceedings of the american mathematical society Volume 106, Number I, May 1989 ON INNER AUTOMORPHISMS OF FINITE GROUPS MARTIN R. PETTET (Communicated by Warren J. Wong) Abstract. It is shown that in certain classes of finite groups, inner automor- phisms are characterized by an extension property and also by a dual lifting property. This is a consequence of the fact that for any finite group G and any prime p , there is a p-group P and a semidirect product H = GP such that P is characteristic in H and every automorphism of H induces an inner automorphism on H/P. Every inner automorphism of a group G possesses the property that it ex- tends to an automorphism of any group which contains G as a subgroup. P. Schupp [2] has shown that in the category of all groups, this extendibility prop- erty characterizes inner automorphisms. However, since Schupp's argument uses a free product construction, the question arises whether inner automor- phisms can be similarly characterized in classes of groups which are not closed under free products. In this note, we confirm that the extendibility property does indeed charac- terize inner automorphisms in certain classes of finite groups. As it happens, with the same stroke the dual question of whether inner automorphisms are precisely those which can be lifted to homomorphic pre-images is also settled (affirmatively) for those classes. The basis for these conclusions amounts to an observation about a remarkable construction of H. Heineken and H. Liebeck [1] (and a subsequent extension of U. H. M. Webb [3]).
    [Show full text]
  • MAS330 FULL NOTES Week 1. Euclidean Algorithm, Linear
    MAS330 FULL NOTES Week 1. Euclidean Algorithm, Linear congruences, Chinese Remainder Theorem Elementary Number Theory studies modular arithmetic (i.e. counting modulo an integer n), primes, integers and equations. This week we review modular arithmetic and Euclidean algorithm. 1. Introduction 1.1. What is Number Theory? Not trying to answer this directly, let us mention some typical questions we will deal with in this course. Q. What is the last decimal digit of 31000? A. The last decimal digit of 31000 is 1. This is a simple computation we'll do using Euler's Theorem in Week 4. Q. Is there a formula for prime numbers? A. We don't know if any such formula exists. P. Fermat thought that all numbers of the form 2n Fn = 2 + 1 are prime, but he was mistaken! We study Fermat primes Fn in Week 6. Q. Are there any positive integers (x; y; z) satisfying x2 + y2 = z2? A. There are plenty of solutions of x2 + y2 = z2, e.g. (x; y; z) = (3; 4; 5); (5; 12; 13). In fact, there are infinitely many of them. Ancient Greeks have written the formula for all the solutions! We'll discover this formula in Week 8. Q. How about xn + yn = zn for n ≥ 3? A. There are no solutions of xn + yn = zn in positive integers! This fact is called Fermat's Last Theorem and we discuss it in Week 8. Q. Which terms of the Fibonacci sequence un = f1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233;::: g are divisible by 3? A.
    [Show full text]
  • Graduate Algebra, Fall 2014 Lecture 5
    Graduate Algebra, Fall 2014 Lecture 5 Andrei Jorza 2014-09-05 1 Group Theory 1.9 Normal subgroups Example 1. Specific examples. 1. The alternating group An = ker " is a normal subgroup of Sn as " is a homomorphism. 2. For R = Q; R or C, SL(n; R) C GL(n; R). 3. Recall that for any group G, Z(G) C G and G=Z(G) is a group, which we'll identify later as the group × of inner automorphisms. If R = Z=pZ; Q; R or C then R In = Z(GL(n; R)) and denote the quotient PGL(n; R) = GL(n; R)=R× The case of SL(n; R) is more subtle as the center is the set of n-th roots of unity in R, which depends on 2 what R is. For example Z(SL(2; R)) = ±I2 but Z(SL(3; R)) = I3 while Z(GL(3; C)) = fI3; ζ3I3; ζ3 I3g. a b 0 1 a b c 0 4. But f g is not normal in GL(2;R). Indeed, if w = then w w = . 0 c 1 0 0 c b a 1 b a b 5. But f g is a normal subgroup of f g. 0 1 0 c Remark 1. If H; K ⊂ G are subgroups such that K is normal in G then HK is a subgroup of G. Indeed, KH = HK. Interlude on the big picture in the theory of finite groups We have seen that if G is a finite group and N is a normal subgroup then G=N is also a group.
    [Show full text]
  • The Purpose of These Notes Is to Explain the Outer Automorphism Of
    The Goal: The purpose of these notes is to explain the outer automorphism of the symmetric group S6 and its connection to the Steiner (5, 6, 12) design and the Mathieu group M12. In view of giving short proofs, we often appeal to a mixture of logic and direct computer-assisted enumerations. Preliminaries: It is an exercise to show that the symmetric group Sn ad- mits an outer automorphism if and only if n = 6. The basic idea is that any automorphism of Sn must map the set of transpositions bijectively to the set of conjugacy classes of a single permutation. A counting argument shows that the only possibility, for n =6 6, is that the automorphism maps the set of transpositions to the set of transpositions. From here it is not to hard to see that the automorphism must be an inner automorphism – i.e., conjugation by some element of Sn. For n = 6 there are 15 transpositions and also 15 triple transpositions, permutations of the form (ab)(cd)(ef) with all letters distinct. This leaves open the possibility of an outer automorphism. The Steiner Design: Let P1(Z/11) denote the projective line over Z/11. The group Γ = SL2(Z/11) consists of those transformations x → (ax + b)/(cx + d) with ad − bc ≡ 1 mod 11. There are 660 = (12 × 11 × 10)/2 elements in this group. Let B∞ = {1, 3, 4, 5, 9, ∞} ⊂ P1(Z/11) be the 6-element subset consisting of the squares mod 11 and the element ∞. Let Γ∞ ⊂ Γ be the stabilizer of B∞.
    [Show full text]
  • Introduction to Number Theory
    Introduction to Number Theory INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: ℕ={0 ,1, 2,} . Integers: ℤ={0 ,±1,±2,} . Definition: Divisor ∣ If a∈ℤ can be writeen as a=bc where b , c∈ℤ , then we say a is divisible by b or, b divides a (denoted b a ), or b is a divisor of a . Definition: Prime We call a number p≥2 prime if its only positive divisors are 1 and p . Definition: Congruent ∣ − If d ≥2 , d ∈ℕ , we say integers a and b are congruent modulo d if d a b and write it a≡bmod d . Examples of Number Theory Questions 1. What are the solutions to a 2b2 =c2 ? s2 −t 2 s2 t 2 Answer: a=s t , b= , c= . 2 2 2. Fundamental Theorem of Arithmetic. Each integer can be written as a product of primes; moreover, the representation is unique up to the order of factors. 3. Theorem (Euclid). There are infinitely many prime numbers. 4. Suppose a , b=1 , i.e. a and d have no common divisors except 1. Are there infinitely many primes ≡ p a mod d ? Equivalently, are there infinitely many prime values of the linear polynomial dxa , x∈ℤ ? Answer: Yes (Dirichlet, 1837). 5. Are there infinitely many primes of the form p=x 21, x∈ℤ ? Not known, expect yes. It is known that there are infinitely many numbers n=x 21 such that n is either prime or has 2 prime factors. 2 2 = ≡ 6. What primes can be written as p=a b ? Answer: If p 2 or p 1 mod 4 .
    [Show full text]
  • Primitive Roots
    Chapter 5 Primitive Roots The name primitive root applies to a number a whose powers can be used to represent a reduced residue system modulo n. Primitive roots are there- fore generators in that sense, and their properties will be very instrumental in subsequent developments of the theory of congruences, especially where exponentiation is involved. 5.1 Orders and Primitive Roots With gcd(a, n) = 1, we know that the sequence a % n, a2 % n, a3 % n, . must eventually reach 1 and make a loop back to the first term. In fact, Euler’s theorem says that the length of this periodicity is at most φ(n). We will define this length, for a given a, and ask whether it can sometimes equal φ(n). Definition. Suppose a and n > 0 are relatively prime. The order of a modulo n is the smallest positive integer k such that ak % n = 1. We denote this quantity by |a|n, or simply |a| when there is no ambiguity. For example, |2|7 = 3 because x = 3 is the smallest positive solution to the congruence 2x ≡ 1 (mod 7). Exercise 5.1. Find these orders. a) |3|7 b) |3|10 c) |5|12 d) |7|24 e) |4|25 Exercise 5.2. Suppose |a| = 6. Find |ak| for k = 2, 3, 4, 5, 6. 43 44 Theory of Numbers From now on we agree that the notation |a|n implicitly assumes the condition gcd(a, n) = 1, for otherwise it makes no sense. In particular, by Euler’s theorem, |a|n ≤ φ(n).
    [Show full text]