Higher criteria for the regularity of a one-dimensional local field

The concept of irregularity of formal modules in one-dimensional local fields is considered. A connection is obtained between the irregularity of all unramified extensions $M/L$ and the ramification index $e_{(L/K)}$ for a sufficiently wide class of formal groups. The notion of s-irregularity for natural s is introduced (generalization of the notion of irregularity to the case of roots $[\pi^s]$), and similar criteria for irregularity are proved for it for the case of generalized and relative formal Lubin-Tate modules.