Monic Irreducible Polynomials Research Articles

Monogenity of iterates of irreducible binomials

One of the oldest problems of algebraic number theory is to find a method to determine if the ring of integers of a number field K is Z [ θ ] for some θ ∈ K ; a field for which the answer to this question is affirmative is referred to as a monogenic field. Suppose f ( x ) = x m − a ∈ Z [ x ] is a monic irreducible polynomial and α n ∈ C is a root of f n ( x ) , the n-fold composition of f. In this article, we prove a necessary and sufficient condition for K n = Q ( α n ) to be monogenic for every n ∈ N and m ≥ 2.

On the ideal discriminant of some relative pure extensions

Let L = K(α) be an extension of a number field K where α satisfies the monic irreducible polynomial P(X) = Xp −a ∈ R[X] of prime degree p and such that a is pth power free in R := OK (the ring of integers of K). The purpose of this paper is to give an explicit formula for the ideal discriminant DL/K of L over K involving only the prime ideals dividing the principal ideals aR and pR. As an illustration, we compute the discriminant DL/K of a family of septic and quintic pure fields over quadratic fields. Hence a slightly simpler computation of discriminant DL/K is obtained.

The On power integral bases of certain pure number fields defined by $x^{120}-m $

Let $K$ be a pure number field with $\alpha$ a complex root of a monic irreducible polynomial $F(x) = x^{120}-m \in \mathbb{Z}[x]$ with $ m \neq \pm 1 $. In this paper, we study the monogenity of $K$. More precisely, we prove that if $m$ is square-free, $m \not \equiv 1\md{4}$, $m \not \equiv \pm 1 \md{9} $, and $\ol{m}\not \in \{ \mp 1, 7, 18 \} \md {25}$, then $K$ is monogenic. On the other hand, if $m \equiv 1\md{4}$, $m \equiv 1 \md{9} $, or $m \equiv 1 \md{25}$, then $K$ is not monogenic. Our results are illustrated by some computational examples.

On common index divisors and monogenity of certain number fields defined by x^{5}+ax+b

Let K = Q(α) be a number field, where α satisfies the monic irreducible polynomial F (x) = x5 + ax + b belonging to Z[x]. The purpose of this paper is to caracterise when a prime p is a common index divisor of K. More precisely, we give explicitly a necessary and sufficient conditions, on a and b for which K is not monogenic. Some useful examples are also given.

CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS

Abstract Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$ , where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$ , $1\leq m<n$ . We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$ . We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$ . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$ , the symmetric group on n letters.

Distribution of constant terms of irreducible polynomials in ℤ_p[x] whose degree is a product of two distinct odd primes

We obtain explicit formulas for the number of monic irreducible polynomials with prescribed constant term and degree $q_1q_2$ over a finite field, where $q_1$ and $q_2$ are distinct odd~primes. These formulas are derived from work done by Yucas. We show that the number of polynomials of a given constant term depends only on whether the constant term is a $q_1$-residue and/or a $q_2$-residue in the underlying field. We further show that as $k$ becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal. This paper continues work done in [1].

ON A CONJECTURE OF LENNY JONES ABOUT CERTAIN MONOGENIC POLYNOMIALS

Abstract Let $K={\mathbb {Q}}(\theta )$ be an algebraic number field with $\theta $ satisfying a monic irreducible polynomial $f(x)$ of degree n over ${\mathbb {Q}}.$ The polynomial $f(x)$ is said to be monogenic if $\{1,\theta ,\ldots ,\theta ^{n-1}\}$ is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones [‘A brief note on some infinite families of monogenic polynomials’, Bull. Aust. Math. Soc.100 (2019), 239–244] conjectured that if $n\ge 3$ , $1\le m\le n-1$ , $\gcd (n,mB)=1$ and A is a prime number, then the polynomial $x^n+A (Bx+1)^m\in {\mathbb {Z}}[x]$ is monogenic if and only if $n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ is square-free. We prove that this conjecture is true.

Some relations between the irreducible polynomials over a finite field and its quadratic extension

In this paper, we establish some relations between irreducible polynomials over a finite field Fq and its quadratic extension Fq2. First we consider a relation between the numbers of irreducible polynomials of a fixed degree over Fq and Fq2, and some relations between self-reciprocal irreducible polynomials over Fq and self-conjugate-reciprocal irreducible polynomials over Fq2. We also obtain formulas for the number and the product of all self-conjugate-reciprocal irreducible monic (SCRIM) polynomials over Fq2.

Polycyclic codes over [formula omitted]: Classification, Hamming distance, and annihilators

Let p be prime integer, s and m positive integers; f(x) be a monic irreducible polynomial over R where R:=Fpm[u]/〈u2〉. This paper gives a complete classification of polycyclic codes over R associated to f(x)ps, and the other hand, when f(x)∈Fpm[x], their Hamming distances and their annihilators are explicitly given.

Computation of Multiplicative Inverses of non-zero Polynomials of GF(73) by Gauss-Jordan method.

The present manuscript deals with the derivation of multiplicative inverses of all non-zero elemental polynomials of the Galois field with respect to the monic irreducible polynomials of degree 3 over the prime field by Gauss-Jordan method as the Extended Euclidean Algorithm (EEA) is not always successful over the prime field , when prime .

Diminution of Extended Euclidean Algorithm for Finding Multiplicative Inverse in Galois Field

This manuscript deals with the theorem on diminution of the Extended Euclidean Algorithm for finding the multiplicative inverse of non-zero elemental polynomials of Galois field with respect to a monic irreducible polynomial over , where is a prime and is any positive integer. This method became successful in finding the multiplicative inverse of all those non-zero polynomials for which the Extended Euclidean Algorithm fails. We find the inverse of all 342 non-zero elemental polynomials of using an irreducible polynomial We also used Cayley Hamilton’s theorem for finding the multiplicative inverse of the non-zero elemental polynomials of a finite field.

Integral Bases and Monogenity of Pure Number Fields with Non-Square Free Parameters up to Degree 9

AbstractLetKbe a pure number field generated by a rootαof a monic irreducible polynomialf(x)=xn−mwithma rational integer and 3≤n≤ 9 an integer. In this paper, we calculate an integral basis of ℤK, and we study the monogenity ofK, extending former results to the case whenmis not necessarily square-free. Collecting and completing the corresponding results in this more general case, our purpose is to provide a parallel to [Gaál, I.—Remete, L.:Power integral bases and monogenity of pure fields,J.Number Theory,173(2017), 129–146], where only square-free values ofmwere considered.

On The Geometric Determination of Extensions of Non-Archimedean Absolute Values

Abstract Let | | be a discrete non-archimedean absolute value of a field K with valuation ring 𝒪, maximal ideal 𝓜 and residue field 𝔽 = 𝒪/𝓜. Let L be a simple finite extension of K generated by a root α of a monic irreducible polynomial F ∈ O[x]. Assume that F ¯ = ϕ ¯ l $\overline F = \overline \varphi ^l$ in 𝔽[x] for some monic polynomial φ ∈ O[x] whose reduction modulo 𝓜 is irreducible, the φ-Newton polygon N φ ¯ ( F ) $N\overline \phi \left( F \right)$ has a single side of negative slope λ, and the residual polynomial R λ(F )(y) has no multiple factors in 𝔽 φ [y]. In this paper, we describe all absolute values of L extending | |. The problem is classical but our approach uses new ideas. Some useful remarks and computational examples are given to highlight some improvements due to our results.

ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Let q be a prime number and f(x)=xqs−axm−b be a monic irreducible polynomial of degree qs having integer coefficients. Let K=ℚ(𝜃) be an algebraic number field with 𝜃 a root of f(x). We give some explicit conditions involving only a,b,m,q,s for which K is not monogenic. As an application, we show that if p is a prime number of the form 32k+1, k∈ℤ and 𝜃 is a root of a monic polynomial f(x)=x2s−32cpx2r−p∈ℤ[x] with s>4,2∤c,s≠5+r, then ℚ(𝜃) is not monogenic.

Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

For every integer k there exists a bound B=B(k) such that if the characteristic polynomial of gin textrm{SL}_n(q) is the product of le k pairwise distinct monic irreducible polynomials over mathbb {F}_q, then every element x of textrm{SL}_n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n=p is prime, if g has order frac{q^p-1}{q-1}, then every non-scalar element x in textrm{SL}_p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

On monogenity of certain pure number fields defined by $$x^{{2}^{u}.3^{v}} - m$$

Let \(K = \mathbb {Q} (\alpha)\) be a pure number field generated by a complex root \(\alpha \) of a monic irreducible polynomial \(F(x) = x^{{2}^{u}.3^{v}} - m\), with \(m \neq \pm 1 \) a square free rational integer, u, and v two positive integers. In this paper, we study the monogenity of K. The cases \(u = 0\) and \(v=0\) have been previously studied by the first author and Ben Yakkou. We prove that if m ≢ 1 (mod 4) and m ≢ \(\pm\)1 (mod 9), then K is monogenic. But if \(m \equiv 1\) (mod 4) or \(m \equiv 1 \) (mod 9) or \(u = 2\) and \(m \equiv -1\) (mod 9), then K is not monogenic. Some illustrating examples are given too.

Bounds for 2-Selmer ranks in terms of seminarrow class groups

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]/{(F(x))}$, which we call the \textit{semi-narrow class group} of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E/K$ is semistable and the discriminant of $F$ is totally negative, then we frequently determine the $2$-Selmer rank of $E$ by computing the root number of $E$ and the $2$-rank of the narrow class group of $L$.

The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree n over the finite field \({\mathbb {F}}_q\) where the coefficients of \(x^{n-1}\) and x vanish for \(n\ge 3\). In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements \(a\in {\mathbb {F}}_{q^n}\) for which Trace\((a)=0\) and Trace\((a^{-1})=0\).

Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients. Asymptotic expression with explicit error bound is derived, which implies that such polynomials of degree 2n always exist provided that the number of prescribed leading coefficients is slightly less than n/4. Exact expressions are also obtained for fields with two or three elements and with up to two prescribed leading coefficients.

On polynomials with roots modulo almost all primes

Call a monic integer polynomial <em>exceptional</em> if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic <em>quadra