For an infinite family of monogenic trinomials $P(X) = X^3\pm 3rbX-b$ in $\mathbb{Z}\lbrack X\rbrack$, arithmetical invariants of the cubic number field $L = \mathbb{Q}(θ)$, generated by a zero $θ$ of $P(X)$, and of its Galois closure $N = L(\sqrt{d(L)})$ are determined. The conductor $f$ of the cyclic cubic relative extension $N/K$, where $K = \mathbb{Q}(\sqrt{d(L)})$ denotes the unique quadratic subfield of $N$, is proved to be of the form $3^eb$ with $e\in\lbrace 1,2\rbrace$, which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in $L$. The number $m$ of non-isomorphic cubic fields $L_1,\ldots,L_m$ sharing a common discriminant $d(L_i) = d(L)$ with $L$ is determined.
For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous examinations of specific trinomial forms in the literature. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.
A polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called \emph{monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Define ${\mathcal F}(x):=x^m+Ax^{m-1}+B$. In this article, we determine sets of conditions on $m$, $A$, and $B$, such that the power-compositional trinomial ${\mathcal F}(x^{p^n})$ is monogenic for all integers $n\ge 0$ and a given prime $p$. Furthermore, we prove the actual existence of infinite families of such trinomials ${\mathcal F}(x)$.
We call a polynomial monogenic if a root $θ$ has the property that $\mathbb{Z}[θ]$ is the full ring of integers in $\mathbb{Q}(θ)$. Consider the two families of trinomials $x^n + ax + b$ and $x^n + cx^{n-1} + d$. For any $n>2$, we show that these families are monogenic infinitely often and give some positive densities in terms of the coefficients. When $n=5$ or 6 and when a certain factor of the discriminant is square-free, we use the Montes algorithm to establish necessary and sufficient conditions for monogeneity, illuminating more general criteria given by Jakhar, Khanduja, and Sangwan usin…
Let $K$ be a number field generated by a root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we study the problem of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree $n=2^r\cdot3^k$ with $r$ and $k$ two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this pape…
For positive integers $a$ and $b$, we let $[U_n]$ be the Lucas sequence of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{ for $n\ge 2$},\] and let $π(m):=π_{(a,b)}(m)$ be the period length of $[U_n]$ modulo the integer $m\ge 2$, where $\gcd(b,m)=1$. We define an \emph{$(a,b)$-Wall-Sun-Sun prime} to be a prime $p$ such that $π(p^2)=π(p)$. When $(a,b)=(1,1)$, such a prime $p$ is referred to simply as a \emph{Wall-Sun-Sun prime}. We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is \emph{monogenic} if $f(x)$ is irreducibl…
We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and \[\{1,θ,θ^2,\ldots, θ^{N-1}\}\] is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Let $k$ be a positive integer, and let $U_n:=U_n(k,-1)$ be the Lucas sequence $\{U_n\}_{n\ge 0}$ of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}.\] A $k$-Wall-Sun-Sun prime is a prime $p$ such that \[U_{π_k(p)}\equiv 0 \pmod{p^2},\] where $π_k(p)$ is the length of the period of $\{U_n\}_{n\ge…
Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of $\Z[þ]$. But if $ \Z[þ]$ is not integrally closed, then Jhorar and Khanduja's results cannot answer on the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of $K$. More precisely, when $\Z_K …
For a number field $K$ defined by a trinomail $F(x) = x^7+ax+b \in \mathbb{Z}[x]$, Jakhar and Kumar gave some necessary conditions on $a$ and $b$, which guarantee the non-monogenity of $K$ \cite{A6}. In this paper, for every prime integer $p$, we characterize when $p$ is a common index divisor of $K$. In particular, if any one of these conditions holds, then $K$ is not monogenic. In such a way our proposed results extend those of Jakhar and Kumar.
We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials.
In this paper, we deal with the problem of monogenity of number fields defined by monic irreducible trinomials $F(x)=x^{12}+ax^m+b\in \mathbb{Z}[x]$ with $1\leq m\leq11$. We give sufficient conditions on $a$, $b$, and $m$ so that the number field $K$ is not monogenic. In particular, for $m=1$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$ and we provide a partial answer to the Problem $22$ of Narkiewicz \cite{Nar} for these number fields. Our results are illustrated by computational examples.
Let O be an order in an algebraic number field K, i.e., a ring with quotient field K which is contained in the ring of integers of K. The order O is called monogenic, if it is of the shape Z[w], i.e., generated over the rational integers by one element. By a result of Győry (1976), the set of w with Z[w]=O is a union of finitely many equivalence classes, where two elements v,w of O are called equivalent if v+w or v-w is a rational integer. An order O is called k times monogenic if there are at least k different equivalence classes of w with Z[w]=O, and precisely k times monogenic if there are …
Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication theory and so on. Permutation binomials and trinomials attract people's interest due to their simple algebraic form and additional extraordinary properties. In this paper, several new classes of permutation binomials and permutation trinomials are constructed. Some of these permutation polynomials are generalizations of known ones.
Permutation polynomials over finite fields are an interesting and constantly active research subject of study for many years. They have important applications in areas of mathematics and engineering. In recent years, permutation binomials and permutation trinomials attract people's interests due to their simple algebraic forms. In this paper, by reversely using Tu's method for the characterization of permutation polynomials with exponents of Niho type, we propose a new method to construct permutation trinomials with coefficients 1. Moreover, we give the explicit compositional inverses of a cla…
Let $f(x)=x^7+ax+b$ be an irreducible polynomial having integer coefficients and $K=\mathbb{Q}(θ)$ be an algebraic number field generated by a root $θ$ of $f(x)$. In the present paper, for every rational prime $p$, our objective is to determine the necessary and sufficient conditions involving only $a,~b$ so that $p$ is a divisor of the index of the field $K$. In particular, we provide sufficient conditions on $a$ and $b$, for which $K$ is non-monogenic. In a special case, we show that if either $8$ divides both $a\pm1$, $b$ or $32$ divides both $a+4$, $b$, then $K$ is non-monogenic. We illust…
Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $σ$ be a generator of $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. In this paper we provide closed formulas for the coefficients of a $σ$-trinomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the dimension of the kernel of $f$ equals its $σ$-degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characteriz…
Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(θ)$ be an algebraic number field defined by a complex root $θ$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The aim of this paper is to provide the necessary and sufficient conditions involving only $a,c$ and $n$ for a given prime $p$ to divide the index of the subgroup $\Z[θ]$ in $\Z_K$. As a consequence, we provide families of monogenic algebraic number fields. Further, when $\Z_K \neq \Z[θ]$, we determine explicitly the index $[\Z_K : \Z[θ]]$ in some cases.
In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and $\mathbb{F}_{5^k}$ are given. We also study permutation quadrinomials of type $Ax^{q(q-1)+1} + Bx^{2(q-1)+1} + Cx^{q} + x$. Our method is based on the investigation of an algebraic curve associated with a {fractional polynomial} over a finite field.
Let $K=\mathbb Q(θ)$ be an algebraic number field with $θ$ a root of an irreducible trinomial $f(x)=x^6+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, for each prime number $p$ we compute the highest power of $p$ dividing the discriminant of $K$ in terms of the prime powers dividing $a,~b$ and discriminant of $f(x)$. An explicit $p$-integral basis of $K$ is also given for each prime $p$ and a method is described to obtain an integral basis of $K$ from these $p$-integral bases which is illustrated with examples.
Let $K=\Q(θ)$ be an algebraic number field with $θ$ a root of an irreducible quadrinomial $f(x) = x^6+ax^m+bx+c\in\Z[x] $ with $m\in\{2,3,4,5\}$. In the present paper, we give some explicit conditions involving only $a,~b,~c$ and $m$ for which $K$ is non-monogenic. In each case, we provide the highest power of a rational prime $p$ dividing index of the field $K$. In particular, we provide a partial answer to the Problem $22$ of Narkiewicz \cite{Nar} for these number fields. Finally, we illustrate our results through examples.
The Index of Composition of polynomials and its applications
Mathematics Division, School of advanced science and languages, VIT Bhopal University（VIT博帕尔大学高等科学与语言学院数学系）；Department of Mathematics, Shanghai University（上海大学数学系）
The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(α)}(z)$ associated with the dynamical zeta functions $ζ_{house(α)}(z)$ of the Rényi--Parry arithmetical dynamical systems, for $α$ an algebraic integer $α$ of house "$house(α)$" greater than 1, (ii) the discovery of lenticuli of poles of $ζ_{house(α)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(α)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(house(α))$ of the Mahler…
In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisenträger-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.
