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A number field is called a Polya field if the module of integer-valued polynomials over its ring of integers has a regular basis. Let L be a field which is a compositum of two quadratic Polya fields. Some questions were raised on Polyaness of L in [7] . Part was solved in [3] and [8] . Here we develop a general strategy allowing us to treat the remaining cases but also to find all these previous results.

In all available papers, on power integral bases of any pure sextic number fields K generated by a complex root α of a monic irreducible polynomial f ( x ) = x 6 − m ∈ Z [ x ] , it was assumed that the rational integer m ≠ ∓ 1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free rational integer is omitted. We start by calculating an integral basis of Z K ; the ring of integers of K. In particular, we characterize when Z K = Z [ α ] , that is when Z K is monogenic and generated by α. We give sufficient conditions on m, which warranty that K is not monogenic. We finish the paper by investigating the case, where m = e 5 and e ≠ ∓ 1 is a square free rational integer.

By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular., Comment: 14 pages

In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $$a^2+b^4$$ a 2 + b 4 . To show this they used the Jacobi symbol to define the spin of Gaussian integers, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $${\mathbb {Z}}[\zeta _{12}]={\mathbb {Z}}[\zeta _3,i]$$ Z [ ζ 12 ] = Z [ ζ 3 , i ] by using the cubic residue character on the Eisenstein integers $${\mathbb {Z}}[\zeta _3]$$ Z [ ζ 3 ] . Our main theorem says that the cubic spin is equidistributed along prime ideals of $${\mathbb {Z}}[\zeta _{12}]$$ Z [ ζ 12 ] . The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $$a^2+b^6$$ a 2 + b 6 on the Eisenstein integers.

A celebrated theorem of Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number $p$, we prove that there is an integral irreducible matrix with spectral radius $p$, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number $p$, there is an irreducible shift of finite type with entropy $\log(p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data., Comment: The referee's suggestions are incorporated; in particular an upper bound for the Perron--Frobenius degree is derived from the main result. See Theorem 1.6. To appear in Ergodic Theory and Dynamical Systems

Let O be the ring of integers of a number field, and let n ≥ 3 . This paper studies bi-interpretability of the ring of integers Z with the special linear group SL n ( O ) , the general linear group GL n ( O ) and the subgroup T n ( O ) of GL n ( O ) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

Final steps are done in proving that the groups SL ⁡ ( n , Z ) \operatorname {SL}(n,\mathbb {Z}) , GL ⁡ ( n , Z ) \operatorname {GL}(n,\mathbb {Z}) and PGL ⁡ ( n , Z ) \operatorname {PGL}(n,\mathbb {Z}) are ( 2 , 3 ) (2,3) -generated if and only if n ≥ 5 n\ge 5 , and PSL ⁡ ( n , Z ) \operatorname {PSL}(n,\mathbb {Z}) is ( 2 , 3 ) (2,3) -generated if and only if n = 2 n=2 or n ≥ 5 n\ge 5 . In particular, the results cover the remaining cases of n = 8 n=8 , …, 12 12 , and 14 14 .

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

In this work we investigate the general relation between the density of a subset of the ring of integers D of a general global field and the Haar measure of its closure in the profinite completion D. We then study a specific family of sets, the preimages of k-free elements (for any given k є N\{0; 1}) via one variable polynomial maps, showing that under some hypotheses their asymptotic density always exists and it is precisely the Haar measure of the closure in D of their set.

Algebraična teorija števil s konstrukcijo končne razredne grupe pove, v kakšni meri enolična faktorizacija v algebraičnih številskih poljih ne drži. Za nekatere izmed njih je značilna enolična dolžina faktorizacij posameznega elementa. Algebraic number theory introduces finite class group in order to determine the extent of factorization’s non-uniqueness in algebraic number fields. Some have a special property of unique factorization length for each element.

Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposium on information theory, ISIT 2020, 2020) defined LRPC codes over the ring of integers modulo a prime power, inspired by the paper of Kamche and Mouaha (IEEE Trans Inf Theory 65(12):7718–7735, 2019) which explored rank metric codes over finite principal ideal rings. In this work, we successfully extend the work of Renner et al. by constructing LRPC codes over the ring $$\mathbb {Z}_{m}$$ Z m which is not a chain ring. We give a decoding algorithm and we study the failure probability of the decoder.

Let K be an imaginary quadratic number field, and $$ \mathcal{O}_K$$ its ring of integers. In this article, we prove the non-existence of Diophantine $$m$$ -tuples in $$\mathcal{O}_K$$ with the property $$D(-1)$$ , for $$m > 36$$ .

LIGHTS OUT! is a game played on a finite, simple graph. The vertices of the graph are the lights, which may be on or off, and the edges of the graph determine how neighboring vertices turn on or off when a vertex is pressed. Given an initial configuration of vertices that are on, the object of the game is to turn all the lights out. The traditional game is played over $\mathbb{Z}_2$, where the vertices are either lit or unlit, but the game can be generalized to $\mathbb{Z}_k$, where the lights have different colors. Previously, the game was investigated on Cartesian product graphs over $\mathbb{Z}_2$. We extend this work to $\mathbb{Z}_k$ and investigate two other fundamental graph products, the direct (or tensor) product and the strong product. We provide conditions for which the direct product graph and the strong product graph are solvable based on the factor graphs, and we do so using both open and closed neighborhood switching over $\mathbb{Z}_k$.

We prove $p$-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a $p$-adic field $K$ or its ring of integers $R$, respectively. In particular, we show that the rigidity results for algebraic functions underlying the so-called Manin-Mumford Conjecture generalize to suitable $p$-adic analytic functions. In the formal setting, this approach leads us to uncover purely $p$-adic Manin-Mumford type results for formal groups not coming from abelian schemes. Moreover, we observe that a version of the Tate-Voloch Conjecture holds in the $p$-adic setting: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the $p$-adic distance.

Let R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

Fix an odd prime $p$. The results in this paper are modeled after work of Hesselholt and Hesselholt–Madsen on the $p$-typical absolute de Rham–Witt complex in mixed characteristic. We have two primary results. The 1st result is an exact sequence that describes the kernel of the restriction map on the de Rham–Witt complex over $A$, where $A$ is the ring of integers in an algebraic extension of $\textbf{Q}_p$ or where $A$ is a $p$-torsion-free perfectoid ring. The 2nd result is a description of the $p$-power torsion (and related objects) in the de Rham–Witt complex over $A$, where $A$ is a $p$-torsion-free perfectoid ring containing a compatible system of $p$-power roots of unity. Both of these results are analogous to the results of Hesselholt and Madsen. Our main contribution is the extension of their results to certain perfectoid rings. We also provide algebraic proofs of these results, whereas the proofs of Hesselholt and Madsen used techniques from topology.

In this paper, we report some interesting results on permutations on $${\mathbb {Z}}_{n}$$ , the ring of integers modulo n, having full differential uniformity. By full differential uniformity of a permutation f on $${\mathbb {Z}}_{n}$$ , we mean that the cardinality of the set $$\{x\in {\mathbb {Z}}_{n}: f(x+a)-f(x)=b\}$$ is exactly n for some $$a,b\in {\mathbb {Z}}_{n}\setminus \{0\}$$ . We give a sufficient condition for an arbitrary map on $${\mathbb {Z}}_{n}$$ to have full differential uniformity. A necessary and sufficient condition for a permutation to have full differential uniformity over the ring of integers modulo n is also given. Further, we propose an upper bound and two lower bounds on permutations with full differential uniformity on $${\mathbb {Z}}_{n}$$ . We prove that these bounds are non-trivial bounds and give the exact number of permutations with full differential uniformity for a certain class of moduli.

(Non-reduced fibers of an arithmetic scheme) For a reduced projective scheme over the ring of integers of a number field, the set of places over which the fibres of the scheme are not reduced is a finite set. We give an explicit upper bound for the product of the norms of places in this set. For this purpose, we introduce a generalization of the notion of height over the adelic ring. We reduce the general case of a scheme of pure dimension to the case of a hypersurface by using the theory of Chow varieties. The case of a hypersurface is then treated with the help of the resultant of the equation of the hypersurface with some partial derivatives of the equation.

Let [Formula: see text] be a number field, [Formula: see text] its ring of integers, [Formula: see text] its classgroup and [Formula: see text] the class number of [Formula: see text]. Let [Formula: see text] be a finite group. Let [Formula: see text] be a maximal [Formula: see text]-order in the semi-simple algebra [Formula: see text] containing [Formula: see text], and [Formula: see text] its locally free classgroup. Let [Formula: see text] and [Formula: see text]. We define the set [Formula: see text] of Galois module classes realizable by the [Formula: see text]th power of the different to be the set of classes [Formula: see text] such that there exists a Galois extension [Formula: see text] with Galois group isomorphic to [Formula: see text] ([Formula: see text]-extension), which is tamely ramified, and for which the class of [Formula: see text] is equal to [Formula: see text], where we clarify that if [Formula: see text], where [Formula: see text], [Formula: see text] is the [Formula: see text]th root of the inverse different [Formula: see text] (respectively, the different [Formula: see text]) if [Formula: see text] (respectively, [Formula: see text]) when it exists. Let [Formula: see text] be a prime number and [Formula: see text] be a primitive [Formula: see text]th root of unity. In this article, we suppose that [Formula: see text] is cyclic of order [Formula: see text] and [Formula: see text] and [Formula: see text] are linearly disjoint. We prove, sometimes under an assumption on [Formula: see text], that [Formula: see text] is a subgroup of [Formula: see text], by an explicit description using a Stickelberger ideal. In addition, for each [Formula: see text], we determine the set of the Steinitz classes of [Formula: see text], [Formula: see text] runs through the tame [Formula: see text]-extensions of [Formula: see text], and prove that it is a subgroup of [Formula: see text], also sometimes under an hypothesis on [Formula: see text].

Let K be a number field defined by an irreducible polynomial F(X) ∈ ℤ[X] and ℤK its ring of integers. For every prime integer p, we give sufficient and necessary conditions on F(X) that guarantee the existence of exactly r prime ideals of ℤK lying above p, where $$\overline F \left( X \right)$$ factors into powers of r monic irreducible polynomials in $${\mathbb{F}_p}\left[ X \right]$$ . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly r prime ideals of ℤK lying above p. We further specify for every prime ideal of ℤK lying above p, the ramification index, the residue degree, and a p-generator.

For a commutative ring R with unity ( $$1\ne 0$$ ), the zero-divisor graph of R is a simple graph with vertices as elements of $$Z(R)^{*}=Z(R)\setminus \{ 0 \}$$ , where Z(R) is the set of zero-divisors of R and two distinct vertices are adjacent whenever their product is zero. An algorithm is presented to create a zero-divisor graph for the ring of Gaussian integers modulo $$2^{n}$$ for $$n\ge 1$$ . The zero-divisor graph $$\Gamma (\mathbb {Z}_{2^{n}}[i])$$ can be expressed as a generalized join graph $$G[G_{1}, \dots , G_{j}]$$ , where $$G_{i}$$ is either a complete graph (including loops) or its complement and G is the compressed zero-divisor graph of $$\mathbb {Z}_{2^{2n}}$$ . Next, we show that the number of isomorphisms between the zero-divisor graphs for the ring of Gaussian integers modulo $$2^{n}$$ and the ring of integers modulo $$2^{2n}$$ is equal to $$\prod _{j=1}^{2(n-1)}2^{j}!$$ .

Introduction: It is conjectured that the cyclic Legendre pairs of odd lengths >1 always exist. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1, and whose periodic autocorrelation function adds up to the constant value −2 (except at the origin). Here G is a finite cyclic group and Z is the ring of integers. These conditions are fundamental and the closely related structure of Hadamard matrices having a two circulant core and double border is incompletely described in literature, which makes its study especially relevant. Purpose: To describe the two-border two-circulant-core construction for Legendre pairs having three new lengths. Results: To construct new Legendre pairs we use the subsets X={x∈G: a(x)=–1} and Y={x∈G: b(x)=–1} of G. There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing thereby the number of undecided cases to 17. In the last section of the paper we list some new examples of cyclic Legendre pairs for lengths v≤123. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information. Programs for search of Hadamard matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms

Given a standard graded polynomial ring over a commutative Noetherian ring $A$, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when $A$ is a field of characteristic zero, and follows from a result of Peskine and Szpiro when $A$ is a field of positive characteristic; our result applies, for example, when $A$ is the ring of integers., Comment: 7 pages

We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field $K$. We apply the balanced HKZ-reduction theory to study the growth of the {\em $g$-invariants} of the ring of integers of $K$. More precisely, for each positive integer $n$, let $\mathcal O$ be the ring of integers of $K$ and $g_{\mathcal O}(n)$ be the smallest integer such that every sum of squares of $n$-ary $\mathcal O$-linear forms must be a sum of $g_{\mathcal O}(n)$ squares of $n$-ary $\mathcal O$-linear forms. We show that when $K$ has class number 1, the growth of $g_{\mathcal O}(n)$ is at most an exponential of $\sqrt{n}$. This extends the recent result obtained by Beli-Chan-Icaza-Liu on the growth of $g_{\mathbb Z}(n)$ and gives the first sub-exponential upper bound for $g_{\mathcal O}(n)$ for rings of integers $\mathcal O$ other than $\mathbb Z$.

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < \mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

In mathematics and computer sciences, the partitioning of a set into two or more disjoint subsets of equal sums is a well-known NP-complete problem, also referred to as partition problem. There are various approaches to overcome this problem for some particular choice of integers. Here, we use quadratic residue graph to determine the possible partitions of positive integers $ m = 2^{\beta}, q^{\beta}, 2^{\beta}q, $ $ 2q^{\beta}, qp, $ where $ p $, $ q $ are odd primes and $ \beta $ is any positive integer. The quadratic residue graph is defined on the set $ Z_{m} = \{\overline{0}, \overline{1}, \cdots, \overline{m-1}\}, $ where $ Z_{m} $ is the ring of residue classes of $ m $, i.e., there is an edge between $ \overline{x}, $ $ \overline{y}\in Z_{m} $ if and only if $ \overline{x}^{2}\equiv \overline{y}^{2}\; (\text{mod}\; m) $. We characterize these graphs in terms of complete graph for some particular classes of $ m $.

In this brief we employ the discrete Arnold’s cat map over the integer ring $\mathbb {Z}_{3^{m}}$ to construct one-dimensional pseudo-chaotic sequences. We analyze their period properties using the properties of the Fibonacci sequence over $\mathbb {Z}_{3^{m}}$ and show that they have twice the period of the sequences generated by the logistic map over $\mathbb {Z}_{3^{m}}$ recently proposed. Moreover, we investigate the pseudo-chaotic properties of the proposed sequences in the context of pseudo-chaos. Finally, these sequences are employed to design a pseudo-random number generator and a statistical analysis with the NIST statistical test suite is performed.

Let f(x) ∈ Z[x] be monic and irreducible over Q, with deg(f) = n. Let K = Q(θ), where f(θ) = 0, and let ZK denote the ring of integers of K. We say f(x) is non-monogenic if � 1, θ, θ2 , . . . , θn−1 is not a basis for ZK. By extending ideas of Ratliff, Rush and Shah, we construct infinite families of non-monogenic trinomials.

Let $\mathbb {Z}_{4}$ be the integer ring of residue classes modulo 4. In this paper, we construct four infinite families of $\mathbb {Z}_{4}$ -codes with one nonzero Lee weight by their generator matrices. Furthermore, we study the linearity of their Gray images and obtain a family of optimal binary codes.

Let $ R $ be a ring with identity. The commuting graph of $ R $ is the graph associated to $ R $ whose vertices are non-central elements in $ R $, and distinct vertices $ A $ and $ B $ are adjacent if and only if $ AB = BA $. In this paper, we completely determine the automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $, where $ \mathbb{Z}_{p^{s}} $ is the ring of integers modulo $ p^{s} $, $ p $ is a prime and $ s $ is a positive integer.