The corestriction of central simple algebras with dubrovin valuation rings

We analyze the corestriction cor L/F (S) of a central simple algebra S over L with respect to a Dubrovin valuation ring A (resp. B i ) of corL/F(S) (resp. S) extending V on F (resp. W i on L) where L is a finite separable extension of F and the W i are the extensions of V to L for 1 ≤ i ≤ k. Under the suitable conditions, we show that for the value group and for the center of residue ring where N(Z( B i )/ F ) is the normal closure of Z( B i ) over F and m i is an integer depending on which roots of unity lie in F and L.