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Polynomial Imaginary Decompositions for Finite Separable Extensions

Adam Grygiel (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials u , . . . , u m - 1 K [ X , . . . , X m - 1 ] such that f ( j = 0 m - 1 ξ j X j ) = j = 0 m - 1 ξ j u j . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then u , . . . , u m - 1 have no common divisor in K [ X , . . . , X m - 1 ] of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several variables....

Positive characteristic analogs of closed polynomials

Piotr Jędrzejewicz (2011)

Open Mathematics

The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.

Prime rational functions

Omar Kihel, Jesse Larone (2015)

Acta Arithmetica

Let f(x) be a complex rational function. We study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we also derive some conditions for the case of complex polynomials.

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