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By Alexander Prestel, Peter Roquette

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Hence L con- a representative a zero By class a 6 R at set. ) denote the p-ramification eL = I , so that becomes a representative index of convex L . We subgroup of vL; have integer m m Let may defined Let we we us = > vp = O is i + e uniquely ke representable with 0 < i < in e the , 0 < form k put em i k P =n Then v~ = m m Hence of the the admits monomials an of form expansion = a° + of this for termined is the tative - Z o our is a O . The of (a - aO of 2 representatives every > the s element a 6 ~L ...

Let care a root be the (If is closed element contains O 6 K be 0 6 L which algebraically field ; this element 2 - v f ' (n) is the field we the contains K = L o_~f elemen~ K may certain whose f(X) a prime a . L > prime polynomial we a root n for vf(~) . ve a prime a root Eisenstein f(X) such of algebraic in over then representative if of importance K is K. L residue . Hence L con- a representative a zero By class a 6 R at set. ) denote the p-ramification eL = I , so that becomes a representative index of convex L .

Hence L' it i < is should since K of implies to K L induced ~ is compatible be an assumed . 10 in the hypotheses special (i) and case (ii) arguments, to t h i s be of special First where field LIK the above EIK finite that every L ~ L' of G a l o i s We have general case where Henselian. This Since L More precisely identify Kh We now have (as v a l u e d LIK of i n f i n i t e being (i) m a y is H e n s e l i a n there with finite its algebraic image / K h I K finite inherited in admits From this n !

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