Factorisability and the arithmetic of wildly ramified Galois extensions
Let be an abelian extension of -adic fields, and let denote the valuation ring of . We study ideals of the valuation ring of as integral representations of the Galois group . Assuming is absolutely unramified we use techniques from the theory of factorisability to investigate which ideals are isomorphic to an -order in the group algebra . We obtain several general and also explicit new results.
On démontre ici un lemme de Hensel pour les opérateurs différentiels. On en déduit un théorème de factorisation pour des opérateurs différentiels à coefficients dans une extension liouvillienne transcendante d’un corps valué. On obtient en particulier un théorème de factorisation pour des opérateurs différentiels à coefficients dans une extension de par un nombre fini d’exponentielles et de logarithmes algébriquement indépendants sur .
Let H be a Krull monoid with infinite class group and such that each divisor class of H contains a prime divisor. We show that for each finite set L of integers ≥2 there exists some h ∈ H such that the following are equivalent: (i) h has a representation for some irreducible elements , (ii) k ∈ L.
Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix having f evaluated at the greatest common divisor of and as its...