Number Of Monic Irreducible Polynomials Research Articles

On 2-superirreducible polynomials over finite fields

We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

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].

Discriminants of fields generated by polynomials of given height

We obtain upper bounds for the number of monic irreducible polynomials over Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{Z}$$\\end{document} of a fixed degree n and a growing height H for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree n and height at most H. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant, improving previous results of I. E. Shparlinski (2010).

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\((a^{-1})=0\).

On the number of irreducible polynomials of special kinds in finite fields

Let $\mathbb{F}_q$ be the finite field of order $q$ and $f(x)$ be an irreducible polynomial of degree $n$ over $\mathbb{F} _q$. For a positive divisor $n_1$ of $n$, define the $n_1$-traces of $f(x)$ to be $\mathrm{Tr}(\alpha; n_1) = \alpha+\alpha^q+\cdots+\alpha^{q^{n_1-1}}$ where $\alpha$'s are the roots of $f(x)$. Let $N_q^*(n; n_1)$ denote the number of monic irreducible polynomials of degree $n$ over $\mathbb{F} _q$ with nozero $n_1$-traces. Ruskey, Miers and Sawada have found the formula for $N_q^*(n; n)$. Based on the properties of linearized polynomials, we obtain the formula for $N_q^*(n; n_1)$ in the general case, including a new proof to the result by Ruskey, Miers and Sawada.

Liminal reciprocity and factorization statistics

Let M d,n (q) denote the number of monic irreducible polynomials in 𝔽 q [x 1 ,x 2 ,...,x n ] of degree d. We show that for a fixed degree d, the sequence M d,n (q) converges coefficientwise to an explicitly determined rational function M d,∞ (q). The limit M d,∞ (q) is related to the classic necklace polynomial M d,1 (q) by an involutive functional equation we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of symmetric group characters as a consequence of liminal reciprocity, giving a liminal analog of a result of Church, Ellenberg, and Farb.

Irreducible polynomials over [formula omitted] with three prescribed coefficients

For any positive integers n≥3 and r≥1, we prove that the number of monic irreducible polynomials of degree n over F2r in which the coefficients of Tn−1, Tn−2 and Tn−3 are prescribed has period 24 as a function of n, after a suitable normalization. A similar result holds over F5r, with the period being 60. We also show that this is a phenomena unique to characteristics 2 and 5. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements an equidistribution result of Katz.

Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Lattice of Subfields of Fqnr

The problem of finding the number of irreducible monic polynomials of degree r over Fqn is considered in this paper. By considering the fact that an irreducible polynomial of degree r over Fqn has a root in a subfield Fqs of Fqnr if and only if (nr/s,r)=1, we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of Fqnr . We also use the lattice of subfields of Fqnr to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of Fqnr.

Irreducible polynomials with several prescribed coefficients

We study the number of monic irreducible polynomials of degree n over Fq having certain preassigned coefficients, where we assume that the constant term (if preassigned) is nonzero. Hansen and Mullen conjectured that for n⩾3, one can always find an irreducible polynomial with any one coefficient preassigned (regardless of the ground field Fq). Their conjecture was established in all but finitely many cases by Wan, and later resolved in full in work of Ham and Mullen. In this note, we present a new, explicit estimate for the number of irreducibles with several preassigned coefficients. One consequence is that for any ϵ>0, and all large enough n depending on ϵ, one can find a degree n monic irreducible with any ⌊(1−ϵ)n⌋ coefficients preassigned (uniformly in the choice of ground field Fq). For the proof, we adapt work of Kátai and Harman on rational primes with preassigned digits.

On certain diagonal equations over finite fields

Let α , β ∈ F q t ∗ and let N t ( α , β ) denote the number of solutions ( x , y ) ∈ F q t ∗ × F q t ∗ of the equation x q − 1 + α y q − 1 = β . Recently, Moisio determined N 2 ( α , β ) and evaluated N 3 ( α , β ) in terms of the number of rational points on a projective cubic curve over F q . We show that N t ( α , β ) can be expressed in terms of the number of monic irreducible polynomials f ∈ F q [ x ] of degree r such that f ( 0 ) = a and f ( 1 ) = b , where r | t and a , b ∈ F q ∗ are related to α , β . Let I r ( a , b ) denote the number of such polynomials. We prove that I r ( a , b ) > 0 when r ⩾ 3 . We also show that N 3 ( α , β ) can be expressed in terms of the number of monic irreducible cubic polynomials over F q with certain prescribed trace and norm.

Irreducible polynomials over finite fields with prescribed trace/prescribed constant term

We obtain an equivalent version of Carlitz's formula for the number of monic irreducible polynomials of degree n and trace γ ≠ 0 over a finite field via an integer recurrence. Similar expressions for the γ = 0 case are also given. We also obtain formulas for the number of monic irreducible polynomials of degree n and prescribed constant term.

The Number of Irreducible Polynomials and Lyndon Words with Given Trace

The trace of a degree n polynomial f(x) over GF(q) is the coefficient of xn-1 . Carlitz [Proc. Amer. Math. Soc., 3 (1952), pp. 693--700] obtained an expression Iq(n,t) for the number of monic irreducible polynomials over GF(q) of degree n and trace t. Using a different approach, we derive a simple explicit expression for Iq(n,t). If t>0, Iq(n,t) = (\sum \mu(d) q^{n/d})/(qn)$, where the sum is over all divisors d of n which are relatively prime to q. This same approach is used to count Lq(n,t), the number of q-ary Lyndon words whose characters sum to t mod q. This number is given by Lq(n,t) = (\sum {\rm gcd}(d,q) \mu(d) q^{n/d})/(qn)$, where the sum is over all divisors d of n for which gcd(d,q)|t. Both results rely on a new form of Möbius inversion.

Applications of the Large Sieve Inequality for [formula omitted]q[T

The purpose of this note is the following: (1) To get an upper bound for the number of monic irreducible polynomials in Fq[T] obtained by changing coefficients of a polynomial in lower degree terms. (2) To generalize the Titchmarsh Linnik divisor problem to polynomial ring Fq[T] and prove an asymptotic formula for this divisor function.

Primitive Elements of Galois Extensions of Finite Fields

As is well known, ${N_q}(n) = (1/n)\sum \nolimits _{d|n} {\mu (d){q^{n/d}}}$ coincides with the number of monic irreducible polynomials of $\operatorname {GF}(q)[X]$ of degree $n$. In this note we discuss the curve $_n{{\text {N}}_X}(n)$ and the solutions of equations $_n{{\text {N}}_X}(n) = b(b \geq n)$. As a corollary of these results, we present a necessary and sufficient arithmetical condition for $R/K$ to have a primitive element.

Primitive elements of Galois extensions of finite fields

As is well known, ${N_q}(n) = (1/n)\sum \nolimits _{d|n} {\mu (d){q^{n/d}}}$ coincides with the number of monic irreducible polynomials of $\operatorname {GF}(q)[X]$ of degree $n$. In this note we discuss the curve $_n{{\text {N}}_X}(n)$ and the solutions of equations $_n{{\text {N}}_X}(n) = b(b \geq n)$. As a corollary of these results, we present a necessary and sufficient arithmetical condition for $R/K$ to have a primitive element.