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Proof. (1)—>(3): We suppose K((X))=L and we consider a sequence [a;}t10 of nonzero elements of D. For any nonnegative integer k, we define cK=(a0ax ■ ■ ■ a*)-1. Then c= 2^°-o ckXkEK [[X]](ZL, so that there is a nonzero element 6= ^k=0bkXk of /?[[A]] such that &c£/?[[A]]. If the power series b has order r, then b/Xr E D [ [X ] ] and (b/Xr) c E D [ [X ] ], so we may assume that r = 0—that is, bo^O. Then if m is a nonnegative integer, the coefficient of Xm in be is

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