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Let p > 3 be a prime number, ζ be a primitive p-th root of unity. Suppose that the Kummer-Vandiver conjecture holds for p , i.e., that p does not divide the class number of Q (ζ + ζ - 1) . Let λ and ν be the Iwasawa invariants of Q (ζ) and put λ = : ∑ i ∈ I λ i and ν = : ∑ i ∈ I ν i in the usual way, where I is the set of odd integers in { 3 , ... , p - 2 } . We prove for each i ∈ I that λ i ≤ 1 , so that the p-primary part of class group of each Q (exp (2 π - 1 / p n)) is determined by the set { (i , ν i) : i ∈ I } . [ABSTRACT FROM AUTHOR]

The study of linear independence of L (k , χ) for a fixed integer k > 1 and varying χ depends critically on the parity of k vis-à-vis χ. This has been investigated by a number of authors for Dirichlet characters χ of a fixed modulus and having the same parity as k. The focal point of this article is to extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus (consequently a single number field lurking in the background). This entails a very careful and hands-on dealing with the arithmetic of compositum of number fields which we undertake in this work. Our results extend earlier works of the first author with Murty-Rath as well as works of Okada, Murty-Saradha and Hamahata. [ABSTRACT FROM AUTHOR]

The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that $$ \begin{align*} \frac{1}{Q}\sum_{Q In other words, under EH, the $\gamma _q /\!\log q$ in these ranges converge to the one point distribution at $1$. This theorem refines and extends a previous result of Ford, Luca and Moree for prime $q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH. [ABSTRACT FROM AUTHOR]

A ring R is said to be clean if each element of R can be written as the sum of a unit and an idempotent. R is said to be weakly clean if each element of R is either a sum or a difference of a unit and an idempotent, and R is said to be feebly clean if every element r can be written as r = u + e 1 − e 2 , where u is a unit and e 1 , e 2 are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl.17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring ℤ (p) [ C q ] is weakly clean and feebly clean, where p , q are distinct primes. In this paper, we consider a more general setting. Let K be an algebraic number field, its ring of integers, ⊂ a nonzero prime ideal and the localization of at . We investigate when the group ring [ G ] is weakly clean and feebly clean, where G is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when K = ℚ (ζ n) is a cyclotomic field or K = ℚ (d) is a quadratic field. [ABSTRACT FROM AUTHOR]

Denote by ε a primitive N th-root of unity. In this paper, we show that the unit cyclotomic multiple zeta values for μ N generate all the cyclotomic multiple zeta values for μ N when N = 2 , 3 , 4. Moreover, the unit cyclotomic multiple zeta values for μ N can be written as Q -linear combinations of (ζ ( 1 ε )) n , (ζ ( 1 ε − 1 )) n and lower depth terms in each weight n when N = 2 , 3 and 4. By detailed analysis of the motivic Galois action, we compute the coefficients of (ζ ( 1 ε )) n , (ζ ( 1 ε − 1 )) n in the above expressions of unit cyclotomic multiple zeta values. [ABSTRACT FROM AUTHOR]

This paper presents an efficient algorithm for computing 11th-power residue symbols in the cyclo-tomic field ℚ (ζ 11 ) , $ \mathbb{Q}\left({{\zeta }_{11}} \right), $ where 11 is a primitive 11th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7th-power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes. [ABSTRACT FROM AUTHOR]

The absolute Galois group of the cyclotomic field K = Q (ζ p) acts on the étale homology of the Fermat curve X of exponent p. We study a Galois cohomology group which is valuable for measuring an obstruction for K -rational points on X. We analyze a 2-nilpotent extension of K which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of K. For p = 3 , we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology which determines it completely. [ABSTRACT FROM AUTHOR]

A Ducci sequence is a sequence of integer $n$ -tuples generated by iterating the map $$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$ Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given $n$. We prove a new upper bound in the case where $n$ is a power of a prime $p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ for which $2$ is a primitive root and the Pellian equation $x^{2}-py^{2}=-4$ has no solutions in odd integers $x$ and $y$. [ABSTRACT FROM AUTHOR]

In this paper, we present the constructions of rotated ℤ 3 r − 1 -lattices, where r is a positive integer, via ℤ -modules of the ring of the integers ℤ [ ζ 3 r + ζ 3 r − 1 ]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. Lower bounds for the minimum product distances of such construction are also presented. [ABSTRACT FROM AUTHOR]

For an odd prime p satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group G Q ( ζ p ) on the homology of the degree p Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve. [ABSTRACT FROM AUTHOR]

Let ζn be a primitive nth root of unity. As is well known, Z[ζn + ζ-1 n] is the ring of integers of [ζn + ζ-1 n]. We give an alternative proof of this fact by using the resultants of modified cyclotomic polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the tame kernel of the cyclotomic field ℚ(ζ5) is trivial, which confirms a conjecture of Browkin. [ABSTRACT FROM AUTHOR]

Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields. [ABSTRACT FROM AUTHOR]

Let K be a composite field of a cyclotomic field k n of odd conductor n ≧ 3 or even one ≧8 with 4 | n and a totally real algebraic extension field F over the rationals Q and both fields k n and F are linearly disjoint over Q to each other. Then the purpose of this paper is to prove that such a relatively totally real extension field K over a cyclotomic field k n has no power integral basis. Each of the composite fields K is also a CM field over the maximal real subfield k n + ⋅ F of K . This result involves the previous work for K = k n ⋅ F of the Eisenstein field k n = k 3 and the maximal real subfields F = k p n + of prime power conductor p n with p ≧ 5 , and an analogue K = k n ⋅ F of cyclotomic fields k n = k 2 m ( m ≧ 3 ) with a totally real algebraic fields F of K = k 4 ⋅ F with a cyclic cubic field F except for k 4 ⋅ k 7 + and k 4 ⋅ k 3 2 + of conductors 28 and 36. [ABSTRACT FROM AUTHOR]

For an unramified odd prime p in a totally real field, we prove that a single Hecke L -value of the field with cyclotomic twist of a sufficiently large p -power conductor, generates the p -power roots of unity. The proof involves technical but crucial ingredients, namely estimations both on the size of an exponential sum over the unit group and on the number of lattice points in a cone with a given congruence. [ABSTRACT FROM AUTHOR]

Recently the authors [] showed that the algebraic integers of the form $${-m+\zeta_k}$$ are bases of a canonical number system of $${\mathbb {Z}[{\zeta_k]}}$$ provided $${m \geqq \phi(k)+1}$$ , where $${\zeta_k}$$ denotes a k-th primitive root of unity and $${\phi}$$ is Euler's totient function. In this paper we are interested in the questions whether two bases $${-m+\zeta_k}$$ and $${-n+\zeta_k}$$ are multiplicatively independent. We show the multiplicative independence in case that 0 < | m− n| < 10 and | m|, | n| > 1. [ABSTRACT FROM AUTHOR]

A system of generators of the minus part of the Stickelberger ideal of a quadratic twist of a cyclotomic field is obtained. [ABSTRACT FROM AUTHOR]

Let k be a field and G be a finite group acting on the rational function field k(xg |g∈G) by k -automorphisms h(xg)=xhg for any g,h∈G. Noether's problem asks whether the invariant field k(G)=k(xg |g∈G)G is rational (i.e. purely transcendental) over k . In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups G . However, even for the cyclic group Cp of prime order p, it is unknown whether there exist infinitely many primes p such that Q(Cp) is rational over Q . Only known 17 primes p for which Q(Cp) is rational over Q are p≤43 and p=61,67,71 . We show that for primes p<20000, Q(Cp) is not (stably) rational over Q except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that Q(Cp) is not (stably) rational over Q for undetermined 28 primes p out of 46 . [ABSTRACT FROM AUTHOR]

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H2 n+1(GL(ℂ)). For a cyclotomic field F we define a canonical set of elements in K3( F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H3(GL(ℂ)) under the Hurewicz map. Applying our formula to these images yields a value V1( F), which coincides with the Borel regulator R1( F) when our set is a basis of K3( F) modulo torsion. For F = ℚ( e2π i/3) a computation of V1( F) has been made based on our techniques. [ABSTRACT FROM AUTHOR]

Let p be a prime number. Let Cp, the cyclic group of order p, permute transitively a set of indeterminates {x1, . . . , xp}. We prove that the invariant field Q(x1, . . . , xp)Cp is rational over Q if and only if the (p - 1)-th cyclotomic field Q(ζp-1) has class number one. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number, Kn = Q(ζpn+1) the pn+1th cyclotomic field and the relative class number of Kn. Fixing an integer d ∈ Z with , we denote by Ln the imaginary quadratic subextension of the imaginary (2, 2)-extension with Ln ≠ Kn. When d < 0, we have . Denote by and the minus parts of the 2-adic Iwasawa lambda invariants of Kn and Ln, respectively. By a theorem of Friedman, these invariants are stable for sufficiently large n. First, under the assumption that is odd for all n ≥ 1, we give a quite explicit version of this result. Second, we show that the assumption is satisfied for all p ≤ 599. Further, using these results, we compute the invariants and with d = -1, -3 for all p ≤ 599 and all n with the help of the computer. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we consider integral trace forms of cyclotomic fields given by the restrictions of trace forms to and we get orthogonal decompositions of symmetric -bilinear modules into rank one and rank two symmetric -bilinear modules in an explicit way. As a result, we can get canonical forms of symmetric -bilinear modules over the ring of p-adic integers. [Copyright &y& Elsevier]

Abstract: By regarding a finite field as a vector space over the prime field with a basis consisting of powers of an element, a Hamming distance is defined on the finite field with respect to the power basis. We consider the existence of isometric homomorphisms between such finite fields, and characterize the isometric embeddings for even characteristic by arithmetical conditions. Moreover, a canonical Hamming metric is defined in a certain infinite dimensional algebraic extension of a finite field. [Copyright &y& Elsevier]

Abstract: We revisit class number formulas, “”-version of Kroneckerʼs limit formulas and Chowla–Selberg formulas for number fields, based on the theory at of zeta functions. The main tool in our investigation is Heckeʼs formula (Corollary to Theorem 1), which represents the zeta function of an algebraic number field K by the integral of Epsteinʼs zeta series summed over . The formulas are derived without residue computations of zeta functions. Theorem 2 for class numbers is quite useful. For Chowla–Selberg formulas, the second gamma function with a character is utilized. [Copyright &y& Elsevier]

We prove that if ρ ≠ 1/2 is a rational number between zero and one, then there is no integer n > 1 such that n tan(πρ) = tan(nπρ). This proves a conjecture due to E. Gutkin which he formulated in connection with mathematical billiards. It also may be viewed as a rigidity result for the circle in the theory of bicycle curves. [ABSTRACT FROM AUTHOR]

In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for ${\rm dim}_{\mathbb{F}_2}{\rm H}^0(G, U_{K}/U_{K}^2)$, where K = ℚ(ζn), G = Gal(K/ℚ), 픽2 = ℤ/2ℤ and UK is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζn) such that ${\rm dim}_{\mathbb{F}_2}{\rm H}^0(G,U_{K}/U_{K}^2)= 1$. Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, J. Number Theory128 (2008) 2687-2694]. In particular, we prove that the unit square problem does not hold for ℚ(ζn) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields ℚ(ζn)v with v | p but not a square in ℚ(ζn), if n has more than three distinct prime factors. [ABSTRACT FROM AUTHOR]

Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In this paper, we provide an efficient procedure for computing 7th power residue symbols. The method employs arithmetic in the field ℚ(ζ), with ζ a primitive 7th root of unity, and its ring of integers ℤ[ζ]. We give an explicit characterization for an element in ℤ[ζ] to be primary, and provide an algorithm for finding primary associates of integers in ℤ[ζ]. Moreover, we formulate explicit forms of the complementary laws to Kummer's 7th degree reciprocity law, and use Lenstra's norm-Euclidean algorithm in the cyclotomic field. [ABSTRACT FROM AUTHOR]

We show that for any integers n ≥≥ 2 and 0 ≤≤ k ≤≤ n, one may find k distinct nth roots of unity whose sum is 0 if and only if both k and n -- k are expressible as linear combinations of prime factors of n with nonnegative coefficients. [ABSTRACT FROM AUTHOR]

Abstract: Let l be an odd prime which satisfies Vandiver''s conjecture, let be an integer, and let where is a primitive root of unity. Let denote the cyclic group of order l. For each j, , there exists an inclusion of Larson orders in : and a corresponding surjection of Hopf–Swan subgroups . For the cases we investigate the structure of various terms in the sequence of Hopf–Swan subgroups including the Swan subgroup . [Copyright &y& Elsevier]

A novel serial-in parallel-out finite field multiplier using redundant representation is proposed. It is shown that the proposed architecture has either a significantly lower complexity and comparable critical path delay or a significantly smaller critical path delay and comparable complexity in comparison to the previously proposed architectures using the same representation. For the class of fields where there exists a type I optimal normal basis, the proposed multiplier compares favorably to the normal basis multipliers. A digit-level version for the new multiplier is also presented in this paper. [ABSTRACT FROM AUTHOR]

Several mathematical frameworks of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find that they are related to the Riemann zeta function and to incomplete Gauss sums, respectively. [ABSTRACT FROM AUTHOR]

This paper derives an algorithm for constructing codes over finite fields of the algebraic integer ring of the cyclotomic field Q(e 2πi/n) modulo irreducible elements with the norm p², where n ∈ {3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84}, where p x n and p is a prime number. [ABSTRACT FROM AUTHOR]

Regarding any finite field as a residue field of the algebraic integer ring of a cyclotomic field, we select a system of representatives in the ring with minimal Manhattan metric, and introduce a Manhattan weight on the finite field. The linear codes over the finite field with the Mannheim weight are discussed. A geometric method to compute the representatives in Gaussian integers is provided. [ABSTRACT FROM AUTHOR]

The RSA cryptosystem is extended to the algebraic field by using ideal theory. In this paper, we describe the generation algorithm of prime ideals, how to select a representative class using an ideal as modulus, and the algorithm to compute a representative element, for cyclotomic fields and quadratic fields. From here, an RSA cryptosystem can be constructed on cyclotomic fields and quadratic fields. To break completely the proposed cryptosystem, when the public key is a product of inert prime numbers, is the same as for the conventional RSA cryptosystem and its security is better against the Håstad attacks. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(8): 19–29, 2000 [ABSTRACT FROM AUTHOR]

In this paper, one subgroup of the multiplicative group of units in the algebraicinteger ring of the cyclotomic field ℚ(e2πi/m) modulo the ideal (qn) has been decomposed into a direct product of cyclic groups, where m ∈ A = {5,7,8,9,11,12,13,15,16,17, 19, 20,21, 24, 25, 27, 28, 32, 33, 35, 36,40,44, 45, 48, 60, 84},q is a given prime integer, and n is a positive integer. The subgroup can be used to obtain a ϕ(m)-dimensional signal space and to construct block codes, over algebraic integers, which are able to correct some error patterns, where ϕ is the Euler function. [ABSTRACT FROM PUBLISHER]

A ring R is said to be clean if each element of R can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring Z (p) [ C n ] is clean, where Z (p) is the localization of the integers at the prime p. In this paper, we consider a more general setting. Let K be an algebraic number field, O K be its ring of integers, and R be a localization of O K at some prime ideal. We investigate when R [ G ] is clean, where G is a finite abelian group, and obtain a complete characterization for such a group ring to be clean for the case when K = Q (ζ n) is a cyclotomic field or K = Q (d) is a quadratic field. [ABSTRACT FROM AUTHOR]

This paper considers rational integer values of Kloosterman sums over finite fields of characteristic p > 3. We shall prove two main results. The first one is a congruence relation satisfied by possible integer values. One consequence is that there are no Kloosterman zeroes in the case of characteristic p > 3, which generalizes recent works by Shparlinski, Moisio, and Lisoněk on this subject. This, in turn, implies that there are no Dillon type bent functions in the case p > 3, thus answering a question posed recently by Helleseth and Kholosha. Our other main result states that the Kloosterman sum obtains an integer value at a point if and only if the same sum lifted to any extension field remains an integer. [ABSTRACT FROM AUTHOR]