Let $p$ be an irregular prime. Let $K=\Q(ζ)$ be the $p$-cyclotomic field.
From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with degree $[M:\Q]=p$ and an explicit formula for the prime decomposition and ramification of the prime number $p$ in the extensions $S/K$, $M/\Q$ and $S/M$. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classical Algebraic Number Theory.
The homogeneous form $Φ_n(X,Y)$ of degree $φ(n)$ which is associated with the cyclotomic polynomial $ϕ_n(X)$ is dubbed a {\it cyclotomic binary form}. A positive integer $m\ge 1$ is said to be {\it representable by a cyclotomic binary form} if there exist integers $n,x,y$ with $n\ge 3$ and $\max\{|x|, |y|\}\ge 2$ such that $Φ_n(x,y)=m$. We prove that the number $a_m$ of such representations of $m$ by a cyclotomic binary form is finite. More precisely, we have $\,φ(n) \le ({2}/ {\log 3})\log m\, $ and $\, \max\{|x|,|y|\} \le ({2}/{\sqrt{3}})\, m^{1/φ(n)}.\,$ We give a description of the asympto…
We consider $\ell$-adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.
On the classification of small cyclotomic integers
Christian-Albrechts University of Kiel（基尔大学）；Indian Institute of Science Education and Research Pune（印度科学教育研究学院浦那分校）；University of California San Diego（加州大学圣地亚哥分校）；Oakland University（奥克兰大学）
令$a(n, k)$为$n$第分圆多项式的第$k$系数。最近，Ji，Li和Moree证明了对任意整数$m\ge1$，$\{a(mn, k)| n, k\in\mathbb{N}\}=\mathbb{Z}$ .本文改进了这一结果，证明了对任意整数$s>t\ge0$，$$\{a(ns+t, k)| n, k\in\mathbb{N}\}=\mathbb{Z}.$$
We investigate the solvability of the Diophantine equation $x^2-my^2=\pm p$ in integers for certain integer $m$ and prime $p$. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number is strictly bigger than $1$.
This note describes a conjecture involving cyclotomic polynomials and some initial thoughts towards a solution. Given positive integers $m,n$, the conjecture is that either $Φ_m(q)\leqslantΦ_n(q)$ or $Φ_m(q)\geqslantΦ_n(q)$ holds for all integers $q\geqslant 2$. Pomerance and Rubinstein-Salzedo proved the conjecture in a preprint called `Cyclotomic coincidences' [arXiv:1903.01962].
For any prime p and any positive integer n, let B(p,n) denote the n-th layer of the cyclotomic Z_p-extension over the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular B(p,n), we provide evidence for the following conjecture: For all primes p and positive integers n, the ideal class group of B(p,n) is trivial.
Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=Φ_{p^m}\left(Φ_{2^n}(x)\right)$, where $Φ_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\mathbb Q$ and that \[\left\{1,θ,θ^2,\ldots,θ^{2^{n-1}p^{m-1}(p-1)-1}\right\}\] is a basis for the ring of integers of $\mathbb Q(θ)$, where $T(θ)=0$.
The exceptional set in Cassel's theorem on small cyclotomic integers
Christian-Albrechts-University of Kiel（基尔大学）；Indian Institute of Science Education and Research, Pune（印度科学教育研究所浦那分校）；University of California San Diego（加州大学圣地亚哥分校）；Florida Atlantic University（佛罗里达大西洋大学）；University of California Santa Barbara（加州大学圣塔芭芭拉分校）；University of Calgary（卡尔加里大学）；Oakland University（奥克兰大学）
We study the set $\mathscr C$ of mean square values of the moduli of the conjugates of cyclotomic integers $β$. For its $k$th derived set $\mathscr C^{(k)}$, we show that $\mathscr C^{(k)}=(k+1)\mathscr C\,\, (k\ge 0)$, so that also ${\mathscr C}^{(k)}+{\mathscr C}^{(\ell)}={\mathscr C}^{(k+\ell+1)}\,\,(k,\ell\ge 0)$. We also calculate the order type of $\mathscr C$, and show that it is the same as that of the set of PV numbers. Furthermore, we describe precisely the restricted set $\mathscr C_p$ where the $β$ are confined to the ring $\mathbb Z[ω_p]$, where $p$ is an odd prime and $ω_p$ is a …
Let $ζ_k$ be a $k$-th primitive root of unity, $m\geqϕ(k)+1$ an integer and $Φ_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+ζ_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|Φ_k(m)|\}$. Moreover we also discuss whether the two bases $-m+ζ_k$ and $-n+ζ_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials that divide their compositions with every positive integer power. We completely characterize such polynomials in terms of their factorizations into cyclotomic polynomials using labeled Hasse diagrams, and construct new integer divisibility sequences based on them. We also show that, unlike the Fibonacci numbers, these non-classical sequences do not have the p…
We propose and justify a generalised approach to prove the polynomial reduction of the RLWE to the PLWE problem attached to the ring of integers of a monogenic number field. We prove such equivalence in the case of the maximal totally real subextension of the 4p-th cyclotomic field, with p arbitrary prime.
We get some results about the factorization of $ϕ_p(M) \in {\mathbb{F}}_2[x]$, where $p$ is a prime number, $ϕ_p$ is the corresponding cyclotomic polynomial and $M$ is a Mersenne prime (polynomial). By the way, we better understand the factorization of the sum of the divisors of $M^{2h}$, for a positive integer $h$.
Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$. Furthermore, we deduce the structure of the $2$-class group of cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.
We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field $\mathbb{Q}(ζ_n)$ are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in $n$. We do so by providing a lower bound, holding for infinitely many positive integers $n$, for the condition number of the Vandermonde matrix of the $n$th cyclotomic polynomial.
We present an algorithm that computes the product of two n-bit integers in O(n log n (4\sqrt 2)^{log^* n}) bit operations. Previously, the best known bound was O(n log n 6^{log^* n}). We also prove that for a fixed prime p, polynomials in F_p[X] of degree n may be multiplied in O(n log n 4^{log^* n}) bit operations; the previous best bound was O(n log n 8^{log^* n}).
Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.
In this paper, we show that if $m$ and $n$ are distinct positive integers and $x$ is a nonzero real number with $Φ_m(x)=Φ_n(x)$, then $\frac{1}{2}<|x|<2$ except when $\{m,n\}=\{2,6\}$ and $x=2$. We also observe that 2 appears to be the largest limit point of the set of values of $x$ for which $Φ_m(x)=Φ_n(x)$ for some $m\neq n$.
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $Φ_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m\ge 1$ we show that every integer occurs as a coefficient of $Φ^*_{mn}(x)$ for some $n\ge 1$. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/Φ_n^*(x),$ the inverse unitary cyclotomic polyno…
An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.
Based on the author's previous work on the Jacobi identity for twisted relative vertex operator algebras and modules and on the generating function identities for affine Lie algebras, we interpret the second difference sequence of the sequence of the number of integer partitions with distinct parts (the strict partitions) as the sequence of the strict partitions of n with at least three parts, the three largest parts consecutive, and the smallest part at least two. The name butterfly describes both the sequence's interpretation and the underlying bijection between the set of strict partitions …
Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number.
We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems that fall in the cyclotomic closure of $\mathbb{K}$. More generally, we consider the finiteness of initial points in the cyclotomic closure for which the orbit contains an algebraic integer of bounded house. This work extends previous results for classical obits generated by one polynomial over $\mathbb{K}$ obtained initially by Dvornicich and Zannier (for prep…
It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. In this article, we propose a method to show that ``cyclotomic polynomials $Φ_{\ell}(X)$ take square free values with positive proportion". Following this method, conditionally, we do prove the Square-free conjecture for all cyclotomic polynomials. The method is applicable to some other class of polynomials as well. Moreover we show t…
Recently the authors showed that the algebraic integers of the form $-m+ζ_k$ are bases of a canonical number system of $\mathbb{Z}[ζ_k]$ provided $m\geq ϕ(k)+1$, where $ζ_k$ denotes a $k$-th primitive root of unity and $ϕ$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+ζ_k$ and $-n+ζ_k$ are multiplicatively independent. We show the multiplicative independence in case that $0<|m-n|<10^6$ and $|m|,|n|> 1$.
Let $\ell$ and $p$ be odd primes. For a positive integer $μ$ let $k_μ$ be the ray class field of $k=\mathbb{Q}(e^{2πi/\ell})$ modulo $2p^μ$. We present certain class fields $K_μ$ of $k$ such that $k_μ\leq K_μ\leq k_{μ+1}$, and find the degree of $K_μ/k_μ$ explicitly. And we also construct, in the sense of Hilbert, primitive generators of the field $K_μ$ over $k_μ$ by using Shimura's reciprocity law and special values of theta constants.
Let $Φ_n(X)$ and $Ψ_n(X)=\frac{X^{n}-1}{Φ_{n}(X)}$ be the $n$-th cyclotomic and inverse cyclotomic polynomials respectively. In this short note, for any pair of divisors $ d_{1} \neq d_{2} $ of $ n $, and integers $l_1$ and $l_2$ such that $ 0 \leq l_{1} \leq φ(d_{1})-1 $ and $ 0 \leq l_{2} \leq φ(d_{2})-1 $, we show that \[\left \langle X^{l_{1}} Ψ_{d_{1}}(X) (1+X^{d_1}+\dots X^{n-d_1}), X^{l_{2}} Ψ_{d_{2}}(X) (1+X^{d_2}+\dots X^{n-d_2}) \right \rangle =0, \] where $ \langle \cdot, \cdot \rangle $ is the inner product on $\mathbb{Q}[X]$ defined by $ \langle \sum_{k} a_{k}X^{k},\sum_{k} b_{k}X…
