On ramification index of composition of complete discrete valuation fields

Abstract

For an extension L/K of discrete valuation fields, let $e_{L/K}$, ${\mathfrak {O}}_{L}$ denote the ramification index and valuation ring of L/K respectively. Let K be a complete discrete valuation field and $L_1/K$, $L_2/K$ be finite linearly disjoint extensions over K. We show that if ${\mathfrak {O}}_{L_1L_2} ={\mathfrak {O}}_{L_1}{\mathfrak {O}}_{L_2}$ or $\mathrm {gcd}(e_{L_1/K}, e_{L_2/K}) =1$, and one of the residue fields $l_1/k,$ $l_2/k$ is separable, then $e_{L_1L_2/L_1} =e_{L_2/K}.$ The analogous results for the residue degrees are also true.