DOCSLIB.ORG

Cyclotomic Fields with Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cyclotomic Fields with Applications Cyclotomic Fields with applications G Eric Moorhouse CYCLOTOMIC FIELDS WITH APPLICATIONS Lecture Notes for Math 5590 Fall 2018 G. Eric Moorhouse University of Wyoming c 2018 ii Preface These notes were written during the summer of 2018, while planning a graduate course for the Fall 2018 semester. The theme was chosen to appeal to students of varying backgrounds, some with interest primarily in number theory, and others more interested in combinatorics, graph theory and finite geometry. As it happens, my research has led me into areas of overlap between these two areas, with cyclotomic fields arising as a common theme. And so beyond the immediate goal of appealing to students with multiple interests, this course was conceived also as a way of crystallizing in my mind some of the finer points of the theory of cyclotomic fields, many of which I had less familiarity with. Students were referred primarily to Washington's book [Wa] for further details on cyclotomic fields, and various other sources as needed. The realities of my teaching environment (perhaps yours too?) mean that I now apportion less lecture time on theory and proofs, with more on examples and applications, than when I first began teaching. In the grand tradition of mathematics, these applications arise largely in ::: (wait for it!) ::: other areas of mathematics. (That may not be strictly true; but as usual, our description of these applications has been rather simplified, sometimes oversimplified, to their mathematical essence, for the sake of brevity.) These applications include algorithms for fast arithmetic with polynomials and integers; constructions and nonexistence results for Hadamard matrices, difference sets, and designs, particularly nets finite affine and projective planes; spectra of Cayley graphs and digraphs over abelian groups; counting solutions to equations over finite fields; the MacWilliams relations for error-correcting codes; Dirichlet's theorem on primes on arithmetic progressions; and mutually unbiased bases (quantum information theory). Given the demands of this pedagogical emphasis, there has been no single reference avail- able where all of these developments can be found. Another design constraint on these notes has been the varying backgrounds of our students, some will have had advanced courses in field theory or number theory, and others not. In order to keep these notes as self-contained as possible, I have included appendices containing many of the results needed from field theory and number theory, omitting the longer proofs; also omitting major results in the theory which to not bear directly upon our particular development or featured applications. I expect that during this fall semester, I will actually summarize much of the content in these appendices during the lectures, rather than leaving students to read these solely on their own. I am indebted to many sources from which I have borrowed extensively, particularly [IR], [LN], [Sa] and [Wa]. Often this has meant rewriting content in my own way, and iii adding details which other authors have left as exercises. I have also looked for ways to avoid explicitly developing all the tools required in some of the standard proofs| not that I feel these tools are unimportant for students to learn, but due to concern that the proliferation of technical definitions and warmup lemmas would overly distract students from the main points. One of my goals, in particular, is a presentation of Gluck's Theorem 14.2 (a beautiful and very accessible argument using cyclotomic integers in a nontrivial and surprising way). Its proof, however, invokes a theorem of Segre usually formulated in the language of projective plane geometry. Not wanting to spend the extra time on such an extended detour for the majority of our students without this conceptual background, I strove instead for an alternative presentation of Segre's Theorem in the language of affine plane geometry. I am very happy with the resulting Theorem 3.14, which I feel is also better adapted to the proof of Gluck's Theorem than the original. I regret omitting several major topics which a more comprehensive textbook would have included: Stickelberger's Theorem, higher reciprocity laws, applications to algebraic coding theory and cryptology, and Bernhard Schmidt's definitive work on the circulant Hadamard conjecture. However, in the spirit of a set of working lecture notes, my priority has been to limit the scope to only what I believe can be accomplished in a single semester. But perhaps in a future revision::: Throughout all my rewriting of standard material, I will certainly have added many of my own errors, for which I take full responsibility. A list of errata will be posted at http://ericmoorhouse.org/courses/5590/ With each mistake/misprint that you encounter in this manuscript, please first check the website to see if it has already been listed; if not, please email me at [email protected] with the necessary correction to add to this list. Thank you! Eric Moorhouse August, 2018 Notational Conventions Throughout these notes, I compose functions right-to-left, as in (στ)(a) = σ(τ(a)). Groups are multiplicative unless otherwise indicated. The symbol ζ denotes a complex root of unity, except when it represents a zeta function (`ala Riemann, Dedekind, Hasse, etc.). Likewise, `i' signifies either an integer index (sometimes a dummy index of summation), p or −1, again depending on the context. So deal with it. iv Contents Preface ::::: ::::::::::::::::::::::::::::::::::::: iii 1. Finite Cyclic Groups :::::::::::::::::::::::::::::: 1 2. Cyclotomic Polynomials:::::::::::::::::::::::::::4 3. Finite Fields :::::::::::::::::::::::::::::::::::::: 8 Squares and Nonsquares 9 Automorphisms of Finite Fields 12 Polynomials versus Functions 13 Counting Irreducible Polynomials 15 Segre's Theorem 16 4. Cyclotomic Fields and Integers:::::::::::::::::::19 5. Fermat's Last Theorem :::::::::::::::::::::::::: 29 6. Characters of Finite Abelian Groups ::::::::::::: 35 Spectra of Cayley Graphs and Digraphs 40 Error-Correcting Codes 41 The Fast Fourier Transform 44 Fast Polynomial Multiplication 46 Fast Integer Multiplication 47 7. Group Rings R[G] ::::::::::::::::::::::::::::::: 49 The Rational Group Algebra of a Finite Cyclic Group 50 Direct Products 52 8. Difference Sets:::::::::::::::::::::::::::::::::::54 9. Hadamard Matrices :::::::::::::::::::::::::::::: 64 Skew-Type Hadamard Matrices 66 Williamson-Hadamard Matrices 67 Regular Hadamard Matrices 73 Circulant Hadamard Matrices 74 10. Quadratic Reciprocity ::::::::::::::::::::::::::: 76 11. Gauss and Jacobi Sums :::::::::::::::::::::::::: 84 12. Zeta Functions and L-Functions::::::::::::::::::91 13. Exponential Sums ::::::::::::::::::::::::::::::: 96 14. Affine Planes ::::::::::::::::::::::::::::::::::: 104 15. Nets::::::::::::::::::::::::::::::::::::::::::::107 16. Mutually Unbiased Bases ::::::::::::::::::::::: 116 17. Weil's Bound ::::::::::::::::::::::::::::::::::: 124 Appendices A1. Fields and Extensions:::::::::::::::::::::::::::129 Matrix Representations of Field Extensions 132 A2. Polynomials and Irreducibility :::::::::::::::::: 136 A3. Algebraic Integers :::::::::::::::::::::::::::::: 138 v A4. Normal and Separable Extensions :: ::::::::::::: 148 A5. Field Automorphisms and Galois Theory::::::::153 A6. Dedekind Zeta Functions and Dirichlet Series ::: 163 A7. Symmetric Polynomials ::::::::::::::::::::::::: 167 A8. Computational Software :::::::::::::::::::::::: 170 PARI/GP 171 Mathematica 172 Bibliography :::::::::::::::::::::::::::::::::::: 175 Index ::::::::::::::::::::::::::::::::::::::::::: 179 vi 1. Finite Cyclic Groups A cyclic group is a group generated by a single element. A cyclic group may be finite or infinite. Every infinite cyclic group is isomorphic to the additive group of Z; or equivalently, the multiplicative subgroup hπi = fπk : k 2 Zg ⊂ C×. (Here C× is the multiplicative group of nonzero complex numbers; and one can replace π by any nonzero complex number which is not a root of unity.) Every finite cyclic group of order n is isomorphic to the additive group Z=nZ of integers mod n; equivalently, the multiplicative subgroup of complex nth roots of unity. The latter group is n 2 n−1 fz 2 C : z = 1g = hζi = f1; ζ; ζ ; : : : ; ζ g × where ζ = ζn is a primitive n-th root of unity, i.e. an element of order n in C . 2πki=n Recall that C contains exactly φ(n) primitive n-th roots of unity e where 1 6 k 6 n, gcd(k; n) = 1 and Euler's totient function φ(n) denotes the number of values of k satisfying the latter conditions. Evidently φ(n) is the number of generators in an arbitrary cyclic group of order n. In the preceding context, we have used the additive cyclic group Z=nZ in which the generators are the elements relatively prime to n. But we will often prefer that our groups be written multiplicatively. Thus in the generic case, an arbitrary cyclic group of order n > 1 may be expressed (up to isomorphism) as a multiplicative group G = hx : xn = 1i = f1; x; x2; : : : ; xn−1g generated by an element x of order n. Theorem 1.1. Let G = hxi be a (multiplicative) cyclic group of order n > 1. Then k G has φ(n) generators x , 1 6 k 6 n, gcd(k; n) = 1. Every subgroup of G is cyclic of order d dividing n. Conversely, for every positive d n, G has a unique subgroup of order d given by hxn=di. Thus X n = φ(d): 16djn Proof. Most of Theorem 1.1 is proved by the Division Algorithm. For example if H is a subgroup of G, let d 2 f1; 2; : : : ; ng be minimal such that xd 2 H. (Note that the set of d 2 f1; 2; : : : ; ng satisfying xd 2 H is nonempty since xn = 1 2 H; so the minimum such d d is defined.) So hx i ⊆ H. We have n = qd + r for some integers q; r with 0 6 r < d. If 0 < r < d then xr = xn(xd)−q 2 H, contradicting the minimality of d; so we must have d n. Now hxdi ⊆ H; and to prove equality, let h 2 H, so h = xj for some j. Again by the 0 0 0 Division Algorithm, j = q d + r where 0 6 r < d; and again using the minimality of d, 0 j d q0 d d P we have r = 0, so x = (x ) 2 hx i.
Load more

Recommended publications
  • QUARTIC CM FIELDS 1. Background the Study of Complex Multiplication
    QUARTIC CM FIELDS WENHAN WANG Abstract. In the article, we describe the basic properties, general and specific properties of CM degree 4 fields, as well as illustrating their connection to the study of genus 2 curves with CM. 1. Background The study of complex multiplication is closely related to the study of curves over finite fields and their Jacobian. Basically speaking, for the case of non-supersingular elliptic curves over finite fields, the endomorphism ring is ring-isomorphic to an order in an imaginary quadratic extension K of Q. The structure of imaginary extensions of Q has beenp thoroughly studied, and the ringsq of integers are simply generated by f1; Dg if D ≡ 1 mod 4, f D g ≡ or by 1; 4 if D 0 mod 4, where D is the discriminant of the field K. The theory of complex multiplication can be carried from elliptic curves to the (Jacobians) of genus 2 (hyperelliptic) curves. More explicitly, the Jacobian of any non-supersingular genus 2 (and hence, hyperelliptic) curve defined over a finite field has CM by an order in a degree 4, or quartic extension over Q, where the extension field K has to be totally imaginary. Description of the endomorphism ring of the Jacobian of a genus 2 curve over a finite field largely depends on the field K for which the curve has CM by. Many articles in the area of the study of genus two curves lead to the study of many properties of the field K. Hence the main goal of this article is, based on the knowledge of the author in the study of the genus 2 curves over finite fields, to give a survey of various, general or specific, properties of degree 4 CM fields.
    [Show full text]
  • The Resultant of the Cyclotomic Polynomials Fm(Ax) and Fn{Bx)
    MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 129 JANUARY 1975, PAGES 1-6 The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn{bx) By Tom M. Apóstol Abstract. The resultant p(Fm(ax), Fn(bx)) is calculated for arbitrary positive integers m and n, and arbitrary nonzero complex numbers a and b. An addendum gives an extended bibliography of work on cyclotomic polynomials published since 1919. 1. Introduction. Let Fn(x) denote the cyclotomic polynomial of degree sp{ri) given by Fn{x)= f['{x - e2"ikl"), k=\ where the ' indicates that k runs through integers relatively prime to the upper index n, and <p{n) is Euler's totient. The resultant p{Fm, Fn) of any two cyclotomic poly- nomials was first calculated by Emma Lehmer [9] in 1930 and later by Diederichsen [21] and the author [64]. It is known that piFm, Fn) = 1 if (m, ri) = 1, m > n>l. This implies that for any integer q the integers Fmiq) and Fn{q) are rela- tively prime if im, ri) = 1. Divisibility properties of cyclotomic polynomials play a role in certain areas of the theory of numbers, such as the distribution of quadratic residues, difference sets, per- fect numbers, Mersenne-type numbers, and primes in residue classes. There has also been considerable interest lately in relations between the integers FJq) and F Jp), where p and q are distinct primes. In particular, Marshall Hall informs me that the Feit-Thompson proof [47] could be shortened by nearly 50 pages if it were known that F Jq) and F Jp) are relatively prime, or even if the smaller of these integers does not divide the larger.
    [Show full text]
  • 08 Cyclotomic Polynomials
    8. Cyclotomic polynomials 8.1 Multiple factors in polynomials 8.2 Cyclotomic polynomials 8.3 Examples 8.4 Finite subgroups of fields 8.5 Infinitude of primes p = 1 mod n 8.6 Worked examples 1. Multiple factors in polynomials There is a simple device to detect repeated occurrence of a factor in a polynomial with coefficients in a field. Let k be a field. For a polynomial n f(x) = cnx + ::: + c1x + c0 [1] with coefficients ci in k, define the (algebraic) derivative Df(x) of f(x) by n−1 n−2 2 Df(x) = ncnx + (n − 1)cn−1x + ::: + 3c3x + 2c2x + c1 Better said, D is by definition a k-linear map D : k[x] −! k[x] defined on the k-basis fxng by D(xn) = nxn−1 [1.0.1] Lemma: For f; g in k[x], D(fg) = Df · g + f · Dg [1] Just as in the calculus of polynomials and rational functions one is able to evaluate all limits algebraically, one can readily prove (without reference to any limit-taking processes) that the notion of derivative given by this formula has the usual properties. 115 116 Cyclotomic polynomials [1.0.2] Remark: Any k-linear map T of a k-algebra R to itself, with the property that T (rs) = T (r) · s + r · T (s) is a k-linear derivation on R. Proof: Granting the k-linearity of T , to prove the derivation property of D is suffices to consider basis elements xm, xn of k[x]. On one hand, D(xm · xn) = Dxm+n = (m + n)xm+n−1 On the other hand, Df · g + f · Dg = mxm−1 · xn + xm · nxn−1 = (m + n)xm+n−1 yielding the product rule for monomials.
    [Show full text]
  • The Kronecker-Weber Theorem
    The Kronecker-Weber Theorem Lucas Culler Introduction The Kronecker-Weber theorem is one of the earliest known results in class field theory. It says: Theorem. (Kronecker-Weber-Hilbert) Every abelian extension of the rational numbers Q is con- tained in a cyclotomic extension. Recall that an abelian extension is a finite field extension K/Q such that the galois group Gal(K/Q) th is abelian, and a cyclotomic extension is an extension of the form Q(ζ), where ζ is an n root of unity. This paper consists of two proofs of the Kronecker-Weber theorem. The first is rather involved, but elementary, and uses the theory of higher ramification groups. The second is a simple application of the main results of class field theory, which classifies abelian extension of an arbitrary number field. An Elementary Proof Now we will present an elementary proof of the Kronecker-Weber theoerem, in the spirit of Hilbert’s original proof. The particular strategy used here is given as a series of exercises in Marcus [1]. Minkowski’s Theorem We first prove a classical result due to Minkowski. Theorem. (Minkowski) Any finite extension of Q has nonzero discriminant. In particular, such an extension is ramified at some prime p ∈ Z. Proof. Let K/Q be a finite extension of degree n, and let A = OK be its ring of integers. Consider the embedding: r s A −→ R ⊕ C x 7→ (σ1(x), ..., σr(x), τ1(x), ..., τs(x)) where the σi are the real embeddings of K and the τi are the complex embeddings, with one embedding chosen from each conjugate pair, so that n = r + 2s.
    [Show full text]
  • 20. Cyclotomic III
    20. Cyclotomic III 20.1 Prime-power cyclotomic polynomials over Q 20.2 Irreducibility of cyclotomic polynomials over Q 20.3 Factoring Φn(x) in Fp[x] with pjn 20.4 Worked examples The main goal is to prove that all cyclotomic polynomials Φn(x) are irreducible in Q[x], and to see what happens to Φn(x) over Fp when pjn. The irreducibility over Q allows us to conclude that the automorphism group of Q(ζn) over Q (with ζn a primitive nth root of unity) is × Aut(Q(ζn)=Q) ≈ (Z=n) by the map a (ζn −! ζn) − a The case of prime-power cyclotomic polynomials in Q[x] needs only Eisenstein's criterion, but the case of general n seems to admit no comparably simple argument. The proof given here uses ideas already in hand, but also an unexpected trick. We will give a different, less elementary, but possibly more natural argument later using p-adic numbers and Dirichlet's theorem on primes in an arithmetic progression. 1. Prime-power cyclotomic polynomials over Q The proof of the following is just a slight generalization of the prime-order case. [1.0.1] Proposition: For p prime and for 1 ≤ e 2 Z the prime-power pe-th cyclotomic polynomial Φpe (x) is irreducible in Q[x]. Proof: Not unexpectedly, we use Eisenstein's criterion to prove that Φpe (x) is irreducible in Z[x], and the invoke Gauss' lemma to be sure that it is irreducible in Q[x]. Specifically, let f(x) = Φpe (x + 1) 269 270 Cyclotomic III p If e = 1, we are in the familiar prime-order case.
    [Show full text]
  • Construction of Regular Polygons a Constructible Regular Polygon Is One That Can Be Constructed with Compass and (Unmarked) Straightedge
    DynamicsOfPolygons.org Construction of regular polygons A constructible regular polygon is one that can be constructed with compass and (unmarked) straightedge. For example the construction on the right below consists of two circles of equal radii. The center of the second circle at B is chosen to lie anywhere on the first circle, so the triangle ABC is equilateral – and hence equiangular. Compass and straightedge constructions date back to Euclid of Alexandria who was born in about 300 B.C. The Greeks developed methods for constructing the regular triangle, square and pentagon, but these were the only „prime‟ regular polygons that they could construct. They also knew how to double the sides of a given polygon or combine two polygons together – as long as the sides were relatively prime, so a regular pentagon could be drawn together with a regular triangle to get a regular 15-gon. Therefore the polygons they could construct were of the form N = 2m3k5j where m is a nonnegative integer and j and k are either 0 or 1. The constructible regular polygons were 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, ... but the only odd polygons in this list are 3,5 and 15. The triangle, pentagon and 15-gon are the only regular polygons with odd sides which the Greeks could construct. If n = p1p2 …pk where the pi are odd primes then n is constructible iff each pi is constructible, so a regular 21-gon can be constructed iff both the triangle and regular 7-gon can be constructed.
    [Show full text]
  • Arxiv:0709.4112V1 [Math.NT] 26 Sep 2007 H Uhri Pnrdb H Okwgnfoundation
    Elle est `atoi cette chanson Toi l’professeur qui sans fa¸con, As ouvert ma petite th`ese Quand mon espoir manquait de braise1. To the memory of Manuel Bronstein CYCLOTOMY PRIMALITY PROOFS AND THEIR CERTIFICATES PREDA MIHAILESCU˘ Abstract. The first efficient general primality proving method was proposed in the year 1980 by Adleman, Pomerance and Rumely and it used Jacobi sums. The method was further developed by H. W. Lenstra Jr. and more of his students and the resulting primality proving algorithms are often referred to under the generic name of Cyclotomy Primality Proving (CPP). In the present paper we give an overview of the theoretical background and implementation specifics of CPP, such as we understand them in the year 2007. Contents 1. Introduction 2 1.1. Some notations 5 2. Galois extensions of rings and cyclotomy 5 2.1. Finding Roots in Cyclotomic Extensions 11 2.2. Finding Roots of Unity and the Lucas – Lehmer Test 12 arXiv:0709.4112v1 [math.NT] 26 Sep 2007 3. GaussandJacobisumsoverCyclotomicExtensionsofRings 14 4. Further Criteria for Existence of Cyclotomic Extensions 17 5. Certification 19 5.1. Computation of Jacobi Sums and their Certification 21 6. Algorithms 23 7. Deterministic primality test 25 8. Asymptotics and run times 29 References 30 1 “Chanson du professeur”, free after G. Brassens Date: Version 2.0 October 26, 2018. The author is sponored by the Volkswagen Foundation. 1 2 PREDA MIHAILESCU˘ 1. Introduction Let n be an integer about which one wishes a decision, whether it is prime or not. The decision may be taken by starting from the definition, thus performing trial division by integers √n or is using some related sieve method, when the decision on a larger set of≤ integers is expected.
    [Show full text]
  • Primality Testing for Beginners
    STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
    [Show full text]
  • Cyclotomic Extensions
    CYCLOTOMIC EXTENSIONS KEITH CONRAD 1. Introduction For any field K, a field K(ζn) where ζn is a root of unity (of order n) is called a cyclotomic extension of K. The term cyclotomic means circle-dividing, and comes from the fact that the nth roots of unity divide a circle into equal parts. We will see that the extensions K(ζn)=K have abelian Galois groups and we will look in particular at cyclotomic extensions of Q and finite fields. There are not many general methods known for constructing abelian extensions of fields; cyclotomic extensions are essentially the only construction that works for all base fields. (Other constructions of abelian extensions are Kummer extensions, Artin-Schreier- Witt extensions, and Carlitz extensions, but these all require special conditions on the base field and thus are not universally available.) We start with an integer n ≥ 1 such that n 6= 0 in K. (That is, K has characteristic 0 and n ≥ 1 is arbitrary or K has characteristic p and n is not divisible by p.) The polynomial Xn − 1 is relatively prime to its deriative nXn−1 6= 0 in K[X], so Xn − 1 is separable over K: it has n different roots in splitting field over K. These roots form a multiplicative group of size n. In C we can write down the nth roots of unity analytically as e2πik=n for 0 ≤ k ≤ n − 1 and see they form a cyclic group with generator e2πi=n. What about the nth roots of unity in other fields? Theorem 1.1.
    [Show full text]
  • Class Group Computations in Number Fields and Applications to Cryptology Alexandre Gélin
    Class group computations in number fields and applications to cryptology Alexandre Gélin To cite this version: Alexandre Gélin. Class group computations in number fields and applications to cryptology. Data Structures and Algorithms [cs.DS]. Université Pierre et Marie Curie - Paris VI, 2017. English. NNT : 2017PA066398. tel-01696470v2 HAL Id: tel-01696470 https://tel.archives-ouvertes.fr/tel-01696470v2 Submitted on 29 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Spécialité Informatique École Doctorale Informatique, Télécommunications et Électronique (Paris) Présentée par Alexandre GÉLIN Pour obtenir le grade de DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE Calcul de Groupes de Classes d’un Corps de Nombres et Applications à la Cryptologie Thèse dirigée par Antoine JOUX et Arjen LENSTRA soutenue le vendredi 22 septembre 2017 après avis des rapporteurs : M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern devant le jury composé de : M. Karim BELABAS Professeur, Université de Bordeaux M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern M.
    [Show full text]
  • Finite Fields: Further Properties
    Chapter 4 Finite fields: further properties 8 Roots of unity in finite fields In this section, we will generalize the concept of roots of unity (well-known for complex numbers) to the finite field setting, by considering the splitting field of the polynomial xn − 1. This has links with irreducible polynomials, and provides an effective way of obtaining primitive elements and hence representing finite fields. Definition 8.1 Let n ∈ N. The splitting field of xn − 1 over a field K is called the nth cyclotomic field over K and denoted by K(n). The roots of xn − 1 in K(n) are called the nth roots of unity over K and the set of all these roots is denoted by E(n). The following result, concerning the properties of E(n), holds for an arbitrary (not just a finite!) field K. Theorem 8.2 Let n ∈ N and K a field of characteristic p (where p may take the value 0 in this theorem). Then (i) If p ∤ n, then E(n) is a cyclic group of order n with respect to multiplication in K(n). (ii) If p | n, write n = mpe with positive integers m and e and p ∤ m. Then K(n) = K(m), E(n) = E(m) and the roots of xn − 1 are the m elements of E(m), each occurring with multiplicity pe. Proof. (i) The n = 1 case is trivial. For n ≥ 2, observe that xn − 1 and its derivative nxn−1 have no common roots; thus xn −1 cannot have multiple roots and hence E(n) has n elements.
    [Show full text]
  • Cyclotomic Extensions
    Chapter 7 Cyclotomic Extensions th A cyclotomic extension Q(ζn) of the rationals is formed by adjoining a primitive n root of unity ζn. In this chapter, we will find an integral basis and calculate the field discriminant. 7.1 Some Preliminary Calculations 7.1.1 The Cyclotomic Polynomial Recall that the cyclotomic polynomial Φn(X) is defined as the product of the terms X −ζ, where ζ ranges over all primitive nth roots of unity in C.Nowannth root of unity is a primitive dth root of unity for some divisor d of n,soXn − 1 is the product of all r cyclotomic polynomials Φd(X) with d a divisor of n. In particular, let n = p be a prime power. Since a divisor of pr is either pr or a divisor of pr−1, we have pr − p − X 1 t 1 p−1 Φ r (X)= − = =1+t + ···+ t p Xpr 1 − 1 t − 1 − pr 1 where t = X .IfX = 1 then t = 1, and it follows that Φpr (1) = p. Until otherwise specified, we assume that n is a prime power pr. 7.1.2 Lemma Let ζ and ζ be primitive (pr)th roots of unity. Then u =(1− ζ)/(1 − ζ) is a unit in Z[ζ], hence in the ring of algebraic integers. Proof. Since ζ is primitive, ζ = ζs for some s (not a multiple of p). It follows that u =(1−ζs)/(1−ζ)=1+ζ+···+ζs−1 ∈ Z[ζ]. By symmetry, (1−ζ)/(1−ζ) ∈ Z[ζ]=Z[ζ], and the result follows.
    [Show full text]