A note on the quotient field of the domain 𝐷[[𝑋]]

Abstract

PROOF. (1) -*(3): We suppose K((X)) =L and we consider a sequence { a, } % of nonzero elements of D. For any nonnegative integer k, we define c,c = (aoa1 ... ak)-1. Then c = _=0 CkXkGK[[X]]CL, so that there is a nonzero element b = EUo bkXk of D [[X]] such that bczD [[X]]. If the power series b has order r, then b/Xr (ED [ [X] ] and (b/Xr)cED [ [X] ], so we may assume that rO -that is, bo-00. Then if m is a nonnegative integer, the coefficient of Xm in bc is

Locations

• Proceedings of the American Mathematical Society - View - PDF