On residual algebraic torsion extensions of a valuation of a field K to K ( x 1 , … , x n )

Abstract

Let v be a valuation of a field K with a value group G v and a residue field k v , w be an extension of v to K(x). Then w is called a residual algebraic torsion extension of v to K(x) if k w /k v is an algebraic extension and G w /G v is a torsion group. In this paper, a residual algebraic torsion extension of v to K(x 1 ,. .. , x n) is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to K(x 1 ,. .. , x n) with given residue field and value group is studied.