Unformatted text preview: Let (M, V) be as in (6.0.4). Then (M, V) has pcurvature zero if and only if the field K contains n = rank M solutions of the ordinary differential equation (6.0.5.1) (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 pdimensional vector space over K p, it follows that if t is a uniformizing parameter in K vor v, the elements 1, t . ... , t pI form a basis of K over K p, and thus provide an isomorphism of KPvector spaces (6.0.6.0) K KP(~ . .. G K p. ptimes In terms of this isomorphism, the vadic topology on K is just the pfold product of the voadic topology on KP; 6*...
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 Fall '11
 NormanKatz
 Vector Space, 0 W, 6.0.4.1, 6.0.4.2, 6.0.4.3, 6.0.5.1

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