Cyclotomic polynomials at roots of unity

Let $\zeta_k$ be a $k$-th primitive root of unity, $m\geq\phi(k)+1$ an integer and $\Phi_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+\zeta_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|\Phi_k(m)|\}$. Moreover we also discuss whether the two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.

Almost all algebra texts define cyclotomic polynomials using primitive nth roots of unity. However, the elementary formula gcd(xm - 1, xn - 1) = xgcd(m,n) - 1 in ℤ[x] can be used to define the cyclotomic polynomials without reference to roots of unity. In this article, partly motivated by cyclotomic polynomials, we prove a factorization property about strong divisibility sequences in a unique factorization domain. After illustrating this property with cyclotomic polynomials and sequences of the form An − Bn, we use the main theorem to prove that the dynamical analogues of cyclotomic polynomials over any unique factorization domain R are indeed polynomials in R[x]. Furthermore, a converse to this article's main theorem provides a simple necessary and sufficient condition for a divisibility sequence to be a rigid divisibility sequence.

Galois polynomials are defined as a generalization of the cyclotomic polynomials. The denition of Galois polynomials (and cyclotomic polynomials) is...

Large absolute values of cyclotomic polynomials at roots of unity

Let (M be a group with unit element e. An element g C (M is a k-th root of unity, for some integer ki ? 2, if gk e. If, in addition, gi j e for all j 1,2, 2 , Ic-i, then g is a primitive k-th root. The purpose of this paper is to show that, when (M is a compact connected Lie group, the structure of primitive roots of unity is, to a considerable extent, determined by the topology of (M and hence is independent of the algebraic structure on (M. Our approach will be, instead of considering only Lie groups, to study a larger class of objects. By an H-manifold we shall mean a triple (M. m, e) where M is a compact connected topological manifold without boundary, e C M, and m: M X M -> M is a map such that mr(x, e) rm(e,x) ==x for all xC M. Define in1: M --VI by m, (x) ==x and, for 7A>2, define Mk(X) m (X, mk-l (X) ). A primitive 1-th root of unity in the H-manifold (M, m, e) is a point x C M such that mk(x) = e but mj (x) 7& e for all j < k. We will studyprimitive roots of unity in this setting. We remark that the subject is of interest, independent of its application to Lie groups. Recent results by Hilton, Belfi [1], Morgan [6], and Zabrodsky produce many new classes of H-manifolds which are not homeomorphic to any Lie group or even, in some cases, of the same homotopy type as any Lie group. We shall show that the existence of primitive roots of unity in an Hmanifold (M, m, e) is independent of the choice of the map m in the sense that:

It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Clq(n|m) such that the latter is a homomorphic image of the quantum superalgebra Uq[osp(2n + 1|2m)]. The Fock space F(n|m) of Clq(n|m) is constructed. For q being a root of unity (q = exp(i?l/k)) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace Fl/k(n|m) of F(n|m). Each Fl/k(n|m) is turned through the above-mentioned homomorphism into an irreducible (root of unity) Uq[osp(2n + 1|2m)] module. For q being a primitive root of unity (l = 1) the corresponding representation is unitary. The module F1/k(n|m) is decomposed into a direct sum of irreducible Uq[sl(m|n)] submodules. The matrix elements of all Cartan?Weyl elements of Uq[sl(m|n)] are given within each such submodule.

In this paper, the so-called compressed Chebyshev polynomials are studied, which are obtained from the original Chebyshev polynomials by suitable scaling transformations. By introducing the concept of cyclotomic pre-polynomials that are in one-to-one correspondence with the well known cyclotomic polynomials, explicit formulas for the irreducible factorization of all four kinds of compressed Chebyshev polynomials are developed. The correspondence between the cyclotomic pre-polynomials and cyclotomic polynomials is based on a polynomial transformation. The same transformation maps all four kinds of the compressed Chebyshev polynomials into polynomials having the property that all their roots are primitive roots of unity, and consequently they are easy to factorize. Generalizations of the applied methods, suitable for the factorization of similar classes of polynomials are also given in the paper.

Let \(N=2n^2-1\) or \(N=n^2+n-1\), for any \(n\ge 2\). Let \(M=\frac{N-1}{2}\). We construct families of prime knots with Jones polynomials \((-1)^M\sum _{k=-M}^{M} (-1)^kt^k\). Such polynomials have Mahler measure equal to 1. If N is prime, these are cyclotomic polynomials \(\Phi _{2N}(t)\), up to some shift in the powers of t. Otherwise, they are products of such polynomials, including \(\Phi _{2N}(t)\). In particular, all roots of unity \(\zeta _{2N}\) occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials.

In the previous papers1-2 we constructed an -dimensional Hopf algebra which is isomorphic to Drinfeld quantum double of Taft's Hopf algebra if and is a primitive nth root of unity, and studied the irreducible representations of . In this paper, we continue our study and examine the irreducible representations of when q is any nth root of 1. We give the structures of all simple -modules, and then classify them. We show that there are only distinct simple -modules up to isomorphism. Though is not a bialgebra, we show that the tensor product of two simple -modules possesses an -module structure in away similar to the case that q is a primitive nth root of 1. Then we give a sufficient and necessary condition for the tensor product of two simple -modules to be semisimple. More generally, we describe the structure of socal of the tensor product. Quantum groups arose from the study of quantum inverse scattering method, especially the Yang-Baxter equation. Quantum groups added new aspects to representation theory. Let be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. It was shown that R can be derived from a left H-module structure and a right H-comodule structure on M for some bialgebra H over k (see[3], [5]. The module and comodule structures satisfy a natural compatibility condition. M together with this structure is called a quantum Yang-Baxter H-module, which is also called a Yetter-Drinfeld module over H (see6-7. A quantum Yang-Baxter H-module is a (left) crossed H-bimodule.[5] Let H be a finite-dimensional Hopf algebra over afield k, and let be the Drinfeld's quantum double derived from H.[8] It is well known9-10 that a vector space M possesses a crossed H-bimodule structure if and only if M possesses a left -module structure. Hence the Yetter-Drinfeld module category is the same as the left -module category Suppose that and that is a primitive nth root of unity. Then is also a primitive nth root of unity. Taft constructed an -dimensional Hopf algebra in[11]. The 's form an interesting class of pointed Hopf algebras from a combinatorial point of view. When n is odd, provides an invariant of three-manifolds.[12] Generally, the double of is of interest in connection with knot theory. Kauffman and Radford showed[13] that is a ribbon Hopf algebra if and only if n is odd. In the paper,[1] the author constructed an infinite-dimensional noncommutative and noncocommutative Hopf algebra for any with . When q is a root of the nth cyclotomic polynomial over Z, has an -dimensional quotient Hopf algebra . If q is a primitive nth root of unity, then is isomorphic to as a Hopf algebra for any . If q is just a nth root of 1, then is merely an algebra.In[2], the author examined the irreducible representations of , equivalently, of , where and q is a primitive nth root of 1. In this paper, we will consider the more general case that q is any nth root of 1, and examine the irreducible representations of the algebra . In this case, let m be the order of q, then , and is an algebra which is not necessarily a bialgebra (Hopf algebra). Let . In Sec. 2, we give the classification of the simple -modules. We show that for any and there exists a unique l-dimensional simple -module , and that for any and , there exists a unique m-dimensional simple -module . We also show that any simple -module is either isomorphic to some or isomorphic to some . Upto isomorphisms, there are only distinct simple -modules. Though is not a bialgebra, we will show that the tensor product of two simple -modules possesses an -module structures which is similar to the case[2] that is a Hopf algebra. In Sec. 3, we consider the tensor product of two simple -modules U and V, and give a sufficient and necessary condition for the tensor product to be semisimple. We first show that if U and V are simple -module with dim or dim then is also a simple -module. Then we prove that is a semisimple -module if and only if , that with is semisimple if and only if , and that is semisimple if and only if or (mod m). Generally, we describe the structure of socal of the tensor product. In particular, if we take , then we can get all the results described in[2].

As in Chapter 3, the nth cyclotomic polynomial Φ n is the minimal polynomial of a primitive nth root of unity. Recall that Φ n is given by $$ {\Phi_n}(z) = \prod\limits_{{\begin{array}{*{20}{c}} {1 \leqslant j \leqslant n} \\ {\gcd \left( {j,n} \right) = 1} \\ \end{array} }} {\left( {z - \exp \left( {{{{j2\pi i}} \left/ {n} \right.}} \right)} \right)} $$ .

Reciprocal relations between cyclotomic fields

We state and prove a mild generalization of Eisenstein’s Criterion for a polynomial to be irreducible, correcting an error that Eisenstein made himself.

This paper presents a new method to test cyclotomic polynomials with Bi-evolution programming based on some existing problems in using traditional methods to test cyclotomic polynomials. The method uses the fact that all the roots moment of a cyclotomic polynomial are roots of unity. Numerical computation results indicate that the algorithm offers an effective way to cyclotomic polynomials, high convergence rate and high accuracy.

Regular systems of weights are certain combinatorial and arithmetic objects related to a generalization of Coxeter elements [S6,7,8 and 11], and introduced in motivation to understand the flat structure for primitive forms for isolated hypersurface singularities [S3] (cf. [Man],[S11]). In the present article, the theory is applied to explain the self-duality of ADE (=simply laced Dynkin diagrams) and the strange duality of Arnold. Beyond the original applications, the study gives further class of dual weight systems, which, for instance, has close connection with Conway group and seems interesting to be studied yet further. On the other hand, the duality of weight systems has an interpretation in terms of certain products of Dedekind eta functions. We give a conjecture on the non-negativity of the Fourier coefficients of the eta-products. The conjecture is solved affirmatively for the cases corresponding to elliptic root systems [S45]. But the meaning is not yet clear. Recently, one finds an equivalence between the duality in the present article and certain string duality in mathematical physics [T]. 0. Introduction. The present article gives a general frame work on the duality of regular systems of weights. For a sake of self-containedness, all proofs are given or sketched except for some basic facts. For simplicity, we shall call a regular system of weights a weight system unless otherwise is stated. A weight system W := (a, b, c;h) is a system of 4 positive integers with some arithmetic constraint (see (1.0)). To W , we attach a cyclotomic polynomial φW , called the characteristic polynomial of the weight system (see (2.1)). The duality we study in the present article appears as a duality between the cyclotomic polynomials. Let us explain this by examples. Let h be a positive integer and let φ(λ) and φ∗(λ) be cyclotomic polynomials whose roots are h-th roots of unity. The polynomials can be decomposed in the form:

The purpose of this note is to establish an identity involving the cyclotomic polynomial and a function of the Ramanujan sums. Some consequences are then derived from this identity.For the reader desiring a background in cyclotomy, (2) is mentioned. Also, (4) is intimately connected with the following discussion and should be consulted.The cyclotomic polynomial Fn(x) is defined as the monic polynomial whose roots are the primitive nth roots of unity. It is well known that2.1For the proof of Corollary 3.2 it is mentioned that Fn(0) = 1 if n &gt; 1 and that Fn(x) &gt; 0 if |x| &lt; 1 and 1 &lt; n.The Ramanujan sums are defined by2.2where the sum is taken over all positive integers r less than or equal to n and relatively prime to n. It is also well known that2.3where the sum is taken over all positive divisors d common to n and k.