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Let Fq be the finite field of q elements, and Fqn its extension of degree n. A normal basis of Fq n over Fq is a basis of the form {α, αq, · · ·, αqn−1 }. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial x n − 1 in Fq, while the latter is closely related to the cyclotomic polynomials. Denote by δ(xn − 1) the set of all distinct monic irreducible factors of xn − 1 in Fq. The criteria for [δ(xn-1)] ≤2 have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for [δ(xn-1)]=s where s is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for [δ(xn-1)]=3,4,5 [ABSTRACT FROM AUTHOR]

Let $ \mathbb{F}_q $ be the finite field of $ q $ elements, and $ \mathbb{F}_{q^{n}} $ its extension of degree $ n $. A normal basis of $ \mathbb{F}_{q^{n}} $ over $ \mathbb{F}_q $ is a basis of the form $ \{\alpha, \alpha^{q}, \cdots, \alpha^{q^{n-1}}\} $. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial $ x^{n}-1 $ in $ \mathbb{F}_q $, while the latter is closely related to the cyclotomic polynomials. Denote by $ \mathfrak{F}(x^{n}-1) $ the set of all distinct monic irreducible factors of $ x^{n}-1 $ in $ \mathbb{F}_q $. The criteria for $ |\mathfrak{F}(x^{n}-1)|\leq 2 $ have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for $ |\mathfrak{F}(x^{n}-1)| = s, $ where $ s $ is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for $ |\mathfrak{F}(x^{n}-1)| = 3, 4, 5. $

Abstract Background The developmental dysplasia of the hip (DDH) can cause a wide range of pathological changes, and often requires surgical treatment. Preoperative evaluation is very important for DDH. We aimed to assess the diagnostic capability of magnetic resonance imaging (MRI) for irreducible aspects preventing hip reduction in DDH. Methods A total of 39 pediatric patients who received DDH evaluation in pediatric orthopedics from January 2015 to December 2019 were included. The samples included 4 cases of bilateral DDH and 35 cases of unilateral DDH, a total of 43 hip joint samples. All patients underwent surgical treatment, pathological examination and MRI of hip joint. Results With pathological results or intraoperative findings as the gold standard, the sensitivity and specificity of MRI were 90.3% and 83.3% for the affected labrum, 92% and 83.3% for thickening of the round ligament, 90.0% and 91.3% for atrophy of the iliopsoas muscle, and 100% and 100% for fibrofatty pulvinar tissue and joint effusion, respectively. Conclutions The MRI showed an extraordinary capability of detecting these irreducible factors and helped surgeon choose the appropriate treatment strategies.

Background: The developmental dysplasia of the hip (DDH) can cause a wide range of pathological changes, and often requires surgical treatment. Preoperative evaluation is very important for DDH. We aimed to assess the diagnostic capability of magnetic resonance imaging (MRI) for irreducible aspects preventing hip reduction in DDH.Methods: A total of 39 pediatric patients who received DDH evaluation in pediatric orthopedics from January 2015 to December 2019 were included. The samples included 4 cases of bilateral DDH and 35 cases of unilateral DDH, a total of 43 hip joint samples. All patients underwent surgical treatment, pathological examination and MRI of hip joint.Results: With pathological results or intraoperative findings as the gold standard, the sensitivity and specificity of MRI were 90.3% and 83.3% for the affected labrum, 92% and 83.3% for thickening of the round ligament, 90.0% and 91.3% for atrophy of the iliopsoas muscle, and 100% and 100% for fibrofatty pulvinar tissue and joint effusion, respectively.Conclutions: The MRI showed an extraordinary capability of detecting these irreducible factors and helped surgeon choose the appropriate treatment strategies. [ABSTRACT FROM AUTHOR]

We discuss a special class of permutation polynomials over finite fields focusing on some recent work on their factorization. In particular we obtain permutation polynomials with various factorization patterns that are favoured for applications. We also address a wide range of problems of current interest concerning irreducible factors of the terms of sequences and iterations of such permutation polynomials. [ABSTRACT FROM AUTHOR]

Let f ( x ) ∈ F q [ x ] be an irreducible polynomial of degree m and exponent e . For each positive integer n , such that ν p ( q − 1 ) ≥ ν p ( e ) + ν p ( n ) for all prime divisors p of n , we show a fast algorithm to determine the irreducible factors of f ( x n ) . Using this algorithm, we give the complete factorization of x n − 1 into irreducible factors in the case where n = d p t , p is an odd prime, q is a generator of the group Z p 2 ⁎ and either d = 2 m with m ≤ ν 2 ( q − 1 ) or d = r a , where r is a prime dividing q − 1 but not p − 1 . [ABSTRACT FROM AUTHOR]

Abstract: This article presents an algorithmic approach to study and compute the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry. [Copyright &y& Elsevier]

Abstract: Swan''s theorem [Pacific J. Math. 12 (1962) 1099–1106] determines the parity of the number of irreducible factors of a binary trinomial. This paper does the same for a binary tetranomial. When phrased in terms of the periodic portion of the factor-parity sequence, the result in several cases is comparable in simplicity to Swan''s result for square-free trinomials. [Copyright &y& Elsevier]

Let F q be the finite field with q elements, and T a positive integer. In this article, we find an asymptotic formula for the total number of monic irreducible binomials in F q x of degree less or equal to T , when T is large enough. We also show explicit lower and upper bounds for the number of binomials in the case when T is small. [ABSTRACT FROM AUTHOR]