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Unformatted text preview: Let (M, V) be as in (6.0.4). Then (M, V) has p-curvature zero if and only if the field K contains n = rank M solutions of the ordinary differential equation (188.8.131.52) (f)= ~ ai (f) i=0 which are linearly independent over the subfield k. K p. (6.0.6) Suppose further that k is a perfect field (then kKP=KP). For any discrete valuation v of K, let v o denote the induced valuation of K p. Clearly the ramification index e(V/Vo) (= the index of the value groups) is p. As K is a p-dimensional vector space over K p, it follows that if t is a uniformizing parameter in K vor v, the elements 1, t . ... , t p-I form a basis of K over K p, and thus provide an isomorphism of KP-vector spaces (184.108.40.206) K--- KP(~ . .. G K p. ptimes In terms of this isomorphism, the v-adic topology on K is just the p-fold product of the vo-adic topology on KP; 6*...
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
- Fall '11