PAC fields over number fields
We prove that if is a number field and is a Galois extension of which is not algebraically closed, then is not PAC over .
Periodic points, multiplicities, and dynamical units.
Pfister's Dimension and the Level of Fields.
Polynomial Imaginary Decompositions for Finite Separable Extensions
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 such that . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then have no common divisor in 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....
Polynomial mappings defined by forms with a common factor
Polynomial subfields of k (x).
Polynomials and convex sequence inequalities.
Polynomials and degrees of maps in real normed algebras
Let be the algebra of quaternions or octonions . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial has a root in . As a consequence, the Jacobian determinant is always non-negative in . Moreover, using the idea of the topological degree we show that a regular polynomial over has also a root in . Finally, utilizing multiplication () in , we prove various results on the topological degree of products...
Polynomials and Vandermonde matrices over the field of quaternions.
Polynomials over solving an embedding problem
The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group , can be embedded in any central extension of if and only if , or and is a sum of two squares. Consequently, for theses values of , every central extension of occurs as a Galois group over .
Polynomials, Symmetry, and Dynamics: an Undertaking in Aesthetics
Polynomials whose values are powers.
Polynomials with nontrivial relations between their roots
Positive characteristic analogs of closed polynomials
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.
Poznámka o jednej vlastnosti dvojprvkového telesa
Prime rational functions
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.
Principal ideal theorems for holomorphy rings in fields.
Product formulas for resultants and Chow forms.
Produkt von Formen.
Proof of the existence of certain triples of polynomials