Irreducible Quartic Polynomials with Factorizations modulo p

Int. J. Math. Math. Sci., 2021

For a Gaussian prime π and a nonzero Gaussian integer β � a + bi ∈ Z[i] with a≥ 1 and |β|≥ 2 + � 2 √ , it was proved that if π � αnβ n + αn− 1β n− 1 + · · · + α1β + α0≕f(β)where n≥ 1, αn ∈ Z[i]\ 0 { }, α0, . . . , αn− 1 belong to a complete residue systemmodulo β, and the digits αn− 1 and αn satisfy certain restrictions, then the polynomial f(x) is irreducible in Z[i][x]. For any quadratic field K ≔ Q( �� m √ ), it is well known that there are explicit representations for a complete residue system in K, but those of the case m ≡ 1 (mod4) are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.

Bulletin of the Polish Academy of Sciences Mathematics, 2004

What should be assumed about the integral polynomials f 1 (x),. .. , f k (x) in order that the solvability of the congruence f 1 (x)f 2 (x) • • • f k (x) ≡ 0 (mod p) for sufficiently large primes p implies the solvability of the equation f 1 (x)f 2 (x) • • • f k (x) = 0 in integers x? We provide some explicit characterizations for the cases when f j (x) are binomials or have cyclic splitting fields.

The Art of Discrete and Applied Mathematics

It is proved that in a finite field F of prime order p, where p is not one of finitely many exceptions, for every polynomial f (x) ∈ F [x] of degree 4 that has a nonzero constant term and is not of the form αg(x) 2 there exists a primitive root β ∈ F such that f (β) is a quadratic residue in F. This refines a result of Madden and Vélez from 1982 about polynomials that represent quadratic residues at primitive roots.

Journal of Number Theory

Fekete polynomials associate with each prime number p a polynomial with coefficients −1 or 1 except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo p. These polynomials were already considered in the 19th century in relation to the studies of Dirichlet L-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomials. In particular, we propose two conjectures on the structure of these Galois groups. We also provide some computational evidence toward the validity of these conjectures.

Acta Arithmetica, 1999

2006

Self-reciprocal irreducible monic (srim) polyn omials over finite fields have been studied in the past. These polynomials can be studied in the context of quad ratic transformation of irreducible polynomials over finite fields. In this talk we present the generalization of some of the results known about srim polynomials to polynomials obtained by quadratic transformation of irreducible polynomials over finite fields. Speaker:Dan Bernstein (University of Illinois at Chicago) Title: Faster factorization into coprimes Abstract: How quickly can we factor a set of univariate polyn mials into coprimes? See http://cr.yp.to/coprimes.html for examples and applications. Bach, Driscoll, and Shallit chieved time n in 1990, wheren is the number of input coefficients; I achieved time n(lg n) in 1995; much more recently I achieved time n(lg n). Speaker:Antonia Bluher (National Security Agency) Title: Hyperquadratic elements of degree 4 Abstract: I will describe joint work with Alain Lasjaunias a ...

Mathematics

Finding irreducible polynomials over Q (or over Z ) is not always easy. However, it is well-known that the mth cyclotomic polynomials are irreducible over Q . In this paper, we define the mth modified cyclotomic polynomials and we get more irreducible polynomials over Q systematically by using the modified cyclotomic polynomials. Since not all modified cyclotomic polynomials are irreducible, a criterion to decide the irreducibility of those polynomials is studied. Also, we count the number of irreducible mth modified cyclotomic polynomials when m = p α with p a prime number and α a positive integer.

Acta Mathematica Hungarica, 2015

In a recent paper , Jankauskas proved some interesting results concerning the reducibility of quadrinomials of the form f (4, x), where f (a, x) = x n +x m +x k +a. He also obtained some examples of reducible quadrinomials f (a, x) with a ∈ Z, such that all the irreducible factors of f (a, x) are of degree ≥ 3.

ITM Web of Conferences, 2018

In order to fully understand the factorization behavior of the ring Int(Z) = { f ∈ Q[x] | f (Z) ⊆ Z} of integer-valued polynomials on Z, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to recognize whether an integer-valued polynomial g d is irreducible in the case where d is a square-free integer and g ∈ Z[x] has fixed divisor d. For integer-valued polynomials with arbitrary composite denominators, so far there is no algorithmic criterion known to recognize whether they are irreducible. We describe a computational method which allows us to recognize all irreducible polynomials in Int(Z). We present some known facts, preliminary new results and open questions.

A polynomial with integer coefficients is given by the ring integers of the integer - valued polynomial is known not to be a unique factorization domain........