INITIAL RAMIFICATION INDEX OF NONINVARIANT VALUATIONS ON FINITE DIMENSIONAL DIVISION ALGEBRAS

Abstract

Let D be a division ring with centre K and dim, D< ? a valuation on K and v a
noninvariant extension of ? to D. We define the initial ramfication index of v over
?, ?(v/ ?) .Let A be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to D with valuation rings A , A ,…, A .
If B= A , it is shown that the following conditions are equivalent: (i) B is a finite A-module, (ii) B is a free A-module, (iii) [B/mB: A/m] = [D: k], (iv) e(v / ?) f(v / ?)= [D: K] and ? (v / ?)= e(v / ?). It is also proved that if ? (v/ ?) = e(v/ ?), and any of (i) - (iv) holds, then v is invariant