Stickelberger-Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over F 2 with degree >1, where A(x) = L(x) + 1 and L(x) = n i=0 x 2 i is a linearized polynomial over F 2 , is reducible except x 2 + x + 1 and x 4 + x + 1. We also give some explicit factors of some special affine pentanomials over F 2 .
Let [Formula: see text] be a finite geometric separable extension of the rational function field [Formula: see text], and let [Formula: see text] be a finite cyclic extension of [Formula: see text] of prime degree [Formula: see text]. Assume that the ideal class number of the integral closure [Formula: see text] of [Formula: see text] in [Formula: see text] is not divisible by [Formula: see text]. Using genus theory and Conner–Hurrelbrink exact hexagon for function fields, we study in this paper the [Formula: see text]-class group of [Formula: see text] (i.e. the Sylow [Formula: see text]-subgroup of the ideal class group of [Formula: see text]) as Galois module, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. The resulting conclusion is used to discuss the relations of class numbers for the biquadratic function fields with their quadratic subfields.
A homogeneous pattern is a linear multivariate polynomial without constant term. Bras-Amorós and García-Sánchez introduced the notion of pattern for numerical semigroups, which generalizes the definition of Arf numerical semigroups. The notion of pattern for numerical semigroups is extended in this paper into a family of homogeneous patterns [Formula: see text]. A numerical semigroup admitting a family of homogeneous patterns [Formula: see text] at [Formula: see text]-level is characterized. We pay our attention in this paper to the families with the action of some permutation groups, especially those consisting of certain partition stabilizers. Stable or sensitive patterns are focused on and we characterize them for several specific permutation groups.
Let [Formula: see text] be a Drinfeld [Formula: see text]-module defined over a global function field [Formula: see text] Let [Formula: see text] be a non-torsion point of [Formula: see text] with infinite [Formula: see text]-orbit. For each [Formula: see text] write the ideal [Formula: see text] as a quotient of relatively prime integral ideals. We establish an analogue of the classical Zsigmondy theorem for the ideal sequence [Formula: see text] i.e. for all but finitely many [Formula: see text] there exists a prime ideal [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]
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