Products of Cyclotomic 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].

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.

The most common use for fast Fourier transform is in high speed signal processing, encryption algorithms and other fields, which use the properties of primitive complex nth root of unity in complex number field. In order to implement the FFT-based large integers multiplication arithmetic whose time complexity is O(nlogn), this paper discusses how to select the primitive nth root of unity for FFT-based large integers multiplication under modulo-p arithmetic, where p is a prime or a composite number, further more, some related proofs which the primitive nth root of unit satisfies the condition of discrete Fourier transform (DFT) and its inverse are given too.

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $\Phi_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $\Phi_n(e^{2\pi i/m})$ with $m\in \{3,4,5,6,8,10,12\}$. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\Phi_n(x)$. We also obtain a formula for $\Phi_n'(e^{2\pi i/m}) / \Phi_n(e^{2\pi i/m})$ with $n \ne m$, which is effectively applied to $m \in \{3,4,6\}$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.

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 > 1 and that Fn(x) > 0 if |x| < 1 and 1 < 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.

The nth cyclotomic polynomial, Φn(z), is the monic polynomial whose φ(n) distinct roots are the nth primitive roots of unity. Φn(z) can be computed efficiently as a quotient of terms of the form (1 - zd) by way of a method the authors call the Sparse Power Series algorithm. We improve on this algorithm in three steps, ultimately deriving a fast, recursive algorithm to calculate Φn(z). The new algorithm, which we have implemented in C, allows us to compute Φn(z) for n > 109 in less than one minute.

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

We would like to prevent, detect, and protect communication and information systems' attacks, which include unauthorized reading of a message of file and traffic analysis or active attacks, such as modification of messages or files, and denial of service by providing cryptographic techniques. If we prove that an encryption algorithm is based on mathematical NP-hard problems, we can prove its security. In this paper, we present a new NTRU-Like public-key cryptosystem with security provably based on the worst-case hardness of the approximate lattice problems (NP-hard problems) in some structured lattices (ideal lattices) in order to attain the applicable objectives of preserving the confidentiality of communication and information system resources (includes hardware, software, firmware, information/data, and telecommunications). Our proposed scheme is an improvement of ETRU cryptosystem. ETRU is an NTRU-Like public-key cryptosystem based on the Eisenstein integers Z [f_3 ] where f_3 is a primitive cube root of unity. ETRU has heuristic security and it has no proof of security. We show that our cryptosystem has security stronger than that of ETRU, over cartesian product of dedekind domains and extended cyclotomic polynomials. We prove the security of our main algorithm from the R-SIS and R-LWE problems as NP-hard problems.

We present a method to deal with the values of polynomials of type [Formula: see text] on the unit circle. We use it to improve the known bounds on various measures of coefficients of cyclotomic and similar polynomials.

A numerical semigroup S is cyclotomic if its semigroup polynomial mathrm {P}_S is a product of cyclotomic polynomials. The number of irreducible factors of mathrm {P}_S (with multiplicity) is the polynomial length ell (S) of S. We show that a cyclotomic numerical semigroup is complete intersection if ell (S)le 2. This establishes a particular case of a conjecture of Ciolan et al. (SIAM J Discrete Math 30(2):650–668, 2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between ell (S) and the embedding dimension of S.

In this paper, the cyclotomic polynomial is generalized and several of its properties based on the Môbius inversion are derived. It is deduced that a polynomial whose roots are the roots of a cyclotomic polynomial multiplied by those of another cyclotomic polynomial is the product of cyclotomic polynomials. Character sums and finite Fourier series are employed for the latter result.

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.

Let a, b be relatively prime integers with |a| > |b| > 0. For any integer n > 0, let π n denote the nth cyclotomic polynomial, denned bywhere ζn is a primitive nth root of unity.

Large absolute values of cyclotomic polynomials at roots of unity

On a problem regarding coefficients of cyclotomic polynomials