By A. Campillo

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**Extra info for Algebroid Curves in Positive Characteristics (Lecture Notes in Mathematics)**

**Example text**

Necessity k primitive. I n the sequel > 0. 14. Conversely, because in k((t)). - Let I--} . T h e n t' basis Corollary if {xi} I~ x. a which = x inseparability index of s T i = i(x) E; k ( ( t ] ) . 1 x. s (x 1 for relative necessary remark has be of the expansion maximal ideal (relative to 24 i ~

With = z J-1 if j as (_D) there is expansion determined by or the for the together no c o n f u s i o n ) . a system conditions: Ig k . k((u)). c J 3) 1 ~< u ( z - - Particutary, (and depend the fixed on the element uniformizing ) < ... < o(z l) r < - - the in expansion it, if u. u(x). + h r-1 set {L(D) there let (D) (O) {x,y} L({ x,y} for }. ). {x,y} If of type + .... UZ expansion (D') one of the following statements true (A) y (B) y 2 I X + a h X T a01 + . . + a 02 X I h + X xh Z I Z Oh ' 1 ~ U(z 1) < O_(x).

D q ' D given of O t is itself a k [ [ x i ) - m o d u l e k((x)] The k['[x]) ideal ideal. s a k [ [ x ) ) - m o d u l e over Let is maximal c 0' [] ' , t h e r e e x i s t of [] ' = m' , i = 1 , 2 . ('~ dim 0' is the ml'=rn ~ iii) 0 ' it maximal ring e x t e n s i o n maximal [] in is of is ii) [] it element k. over Proof: -m 2 is be noetherian. iii) curve an has El' ii 0' x since q' El' ~ q20' . . is ,considered as ~ . '~q nO' This is ~.. e. by q'-adic noetherian, it topology. x)) , then It field.