The field of definition of function fields and a problem in the reducibility of polynomials in two variables
Introduction Let K be a number field, O the ring of integers of K. Suppose we are given a projective curve Z, and a morphism :Z pr(C) (where pr(C) is the projective line) such that and Z are both defined over K. We denote by K(Z) the field of functions of Z defined over K, and from this data we obtain a permutation representation T of the Galois group G(K(Z)^/K(Pr(C))) (where K(Z) is the normal closure of K(Z) over K(Pr(C) ). In Section 1 we investigate (for our needs) the combinatorial and group theoretical aspects of the situation where
Relating Two Genus 0 Problems of John Thompson
Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem 0 g=0 2. We call the exceptional groups 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynomial 0-sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 ℚ cover of the j-line. The latest Nielsen class techniques apply to these dessins d’enfant to see their subtle arithmetic and interesting cusps.
Net Spaces in Categorical Topology a
In the paper we introduce the notion of S‐nets, S a nonempty construct, as a generalization of the usual nets. S‐net spaces and continuous maps between them are then defined in a natural way. For S‐net spaces we introduce a number of axioms and study the categories obtained. We find some topological and Cartesian closed categories of S‐net spaces and describe their relations to certain known topological categories.