Degree of Irregularity and Regular Formal Modules in Local Fields

Abstract

The variation in the irregularity degree of a finite unramified local field extensions of a local field is investigated with respect to a polynomial formal group and in the multiplicative case. The necessary and sufficient conditions for the existence of the psth primitive roots of the psth power of 1 and (endomorphism $${{[{{p}^{s}}]}_{{{{F}_{m}}}}}$$ ) in the Lth unramified extension of the local field K (for all positive integers s) are found. The conditions depend only on the ramification index of the maximal Abelian subextension of the field K Ka/ $${{\mathbb{Q}}_{p}}$$ .