DOCSLIB.ORG

Section 13.6. Let Ζn Be a Primitive N-Th Root of Unity, I.E. Any Generator of the Group of Roots of Unity. the Goal of This

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Section 13.6. Let Ζn Be a Primitive N-Th Root of Unity, I.E. Any Generator of the Group of Roots of Unity. the Goal of This Section 13.6. Let ζn be a primitive n-th root of unity, i.e. any generator of the group of roots of unity. The goal of this lecture is to prove that [Q(ζn): Q] = φ(n); where φ(n) is the Euler's function, that is equal to the number of positive integers k < n such that (k; n) = 1. Let us denote the group of n-th roots of unity by µn and recall that µn ' Z=nZ. Note that µd ⊂ µn () d j n: Indeed, if d j n and ζ is a d-th root of unity, that is ζd = 1 then ζn = 1. Conversely, if ζ is a primitive d-th root of unity and ζ 2 µn, the order of ζ is d, but its order has to divide the order of µn which is n. We know define cyclotomic polynomials which will be minimal polynomials for primitive roots of unity. Definition 1. The n-th cyclotomic polynomial Φn(x) is defined by Y Φn(x) = (x − ζ); where the product is taken over all primitive n-th roots of 1. Note that Y xn − 1 = (x − ζ): ζn=1 If we group together factors (x − ζ) where the order of ζ is d, that is ζ is a primitive d-th root of unity, we get n Y x − 1 = Φd(x): (∗) djn Comparing the degrees of polynomials in both sides of the equation we derive the following interesting equality: X n = φ(n): djn Equality (∗) allows us to compute polynomials Φn(x) by induction on n, for example, Φ1(x) = x − 1 and 2 x − 1 = Φ1(x)Φ2(x) = (x − 1)Φ2(x); hence Φ2(x) = x + 1. Similarly, 3 x − 1 = Φ1(x)Φ3(x) = (x − 1)Φ3(x); 2 hence Φ3(x) = x + x + 1, 4 x − 1 = Φ1(x)Φ2(x)Φ4(x) = (x − 1)(x + 1)Φ4(x); 2 hence Φ4(x) = x + 1, and so on. To show that the degree [Q(ζn): Q] of the cyclotomic extension Q(ζn) over Q is φ(n), it is enough to show that Φn(x) is an irreducible polynomial in Q[x] of degree φ(n). Theorem 2. The cyclotomic polynomial Φn(x) is an irreducible monic polynomial in Z[x] of degree φ(n). 1 2 Proof. First of all, it is obvious from the definition that Φn(x) is a monic polynomial of degree φ(n). We shall show that Φn(x) 2 Z[x] by induction. Clearly, Φ1(x) 2 Z[x], assume Φd(x) 2 Z[x] for all 1 6 d < n. Then n Y x − 1 = f(x)Φn(x) where f(x) = Φd(x): d j n 1<d<n By inductive assumption, f(x) is a monic polynomial with integer coefficients. We now that n n f(x) divides x −1 in Q(ζn)[x], but since both f(x) and x −1 have coefficients in the field Q and Euclidean algorithm does not depend on field extensions, we conclude that f(x) divides n n x − 1 in Q. Now, by Gauss' Lemma we have that f(x) divides x − 1 over Z, which proves that Φn(x) 2 Z[x]. It remains to show that Φn(x) is irreducible. Let us write Φn(x) = f(x)g(x); where f(x) is irreducible. Let ζ be a primitive root of 1, such that f(ζ) = 0, and p be a prime which does not divide n. Then, ζp is again a primitive root of 1, and hence is either a root of f(x) or a root of g(x). Assume that g(ζp) = 0, then ζ is a root of g(xp), and f(x) j g(xp) as f(x) is a minimal polynomial of ζ. Then, g(xp) = f(x)h(x) p p for some h(x) 2 Z[x]. Let us consider the above equality in Fp[x]. There we haveg ¯(x ) =g ¯(x) , and g¯(x)p = f¯(x)h¯(x) ¯ thereforeg ¯(x) and f(x) have a common factor in Fp[x]. This in turn implies that the poly- ¯ ¯ n nomial Φn(x) = f(x)¯g(x) has a multiple root over F, therefore, so does x − 1. But as we know, xn − 1 is separable over any field whose characteristic does not divide n. Thus, for any root ζ of f(x) and any integer a co-prime with n, ζa is also a root of f(x). But the latter implies that any primitive n-th root of 1 is a root of f(x), hence Φn(x) = f(x), which shows that Φn(x) is irreducible. Corollary 3. The degree of the cyclotomic field Q(ζn) over Q equals [Q(ζn): Q] = φ(n):.
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]
  • 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.
    [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]
  • Number Theory and Graph Theory Chapter 3 Arithmetic Functions And
    1 Number Theory and Graph Theory Chapter 3 Arithmetic functions and roots of unity By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: [email protected] 2 Module-4: nth roots of unity Objectives • Properties of nth roots of unity and primitive nth roots of unity. • Properties of Cyclotomic polynomials. Definition 1. Let n 2 N. Then, a complex number z is called 1. an nth root of unity if it satisfies the equation xn = 1, i.e., zn = 1. 2. a primitive nth root of unity if n is the smallest positive integer for which zn = 1. That is, zn = 1 but zk 6= 1 for any k;1 ≤ k ≤ n − 1. 2pi zn = exp( n ) is a primitive n-th root of unity. k • Note that zn , for 0 ≤ k ≤ n − 1, are the n distinct n-th roots of unity. • The nth roots of unity are located on the unit circle of the complex plane, and in that plane they form the vertices of an n-sided regular polygon with one vertex at (1;0) and centered at the origin. The following points are collected from the article Cyclotomy and cyclotomic polynomials by B.Sury, Resonance, 1999. 1. Cyclotomy - literally circle-cutting - was a puzzle begun more than 2000 years ago by the Greek geometers. In their pastime, they used two implements - a ruler to draw straight lines and a compass to draw circles. 2. The problem of cyclotomy was to divide the circumference of a circle into n equal parts using only these two implements.
    [Show full text]
  • Monic Polynomials in $ Z [X] $ with Roots in the Unit Disc
    MONIC POLYNOMIALS IN Z[x] WITH ROOTS IN THE UNIT DISC Pantelis A. Damianou A Theorem of Kronecker This note is motivated by an old result of Kronecker on monic polynomials with integer coefficients having all their roots in the unit disc. We call such polynomials Kronecker polynomials for short. Let k(n) denote the number of Kronecker polynomials of degree n. We describe a canonical form for such polynomials and use it to determine the sequence k(n), for small values of n. The first step is to show that the number of Kronecker polynomials of degree n is finite. This fact is included in the following theorem due to Kronecker [6]. See also [5] for a more accessible proof. The theorem actually gives more: the non-zero roots of such polynomials are on the boundary of the unit disc. We use this fact later on to show that these polynomials are essentially products of cyclotomic polynomials. Theorem 1 Let λ =06 be a root of a monic polynomial f(z) with integer coefficients. If all the roots of f(z) are in the unit disc {z ∈ C ||z| ≤ 1}, then |λ| =1. Proof Let n = degf. The set of all monic polynomials of degree n with integer coefficients having all their roots in the unit disc is finite. To see this, we write n n n 1 n 2 z + an 1z − + an 2z − + ··· + a0 = (z − zj) , − − j Y=1 where aj ∈ Z and zj are the roots of the polynomial. Using the fact that |zj| ≤ 1 we have: n |an 1| = |z1 + z2 + ··· + zn| ≤ n = − 1 n |an 2| = | zjzk| ≤ − j,k 2! .
    [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]