On residual algebraic torsion extensions of a valuation of a field K to K(x1,…,xn)

Let v be a valuation of a field K with a value group Gv and a residue field kv, w be an extension of v to K(x). Then w is called a residual algebraic torsion extension of v to K(x) if kw/kv is an algebraic extension and Gw/Gv is a torsion group. In this paper, a residual algebraic torsion extension of v to K(x1,…,xn) is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to K(x1,…,xn) with given residue field and value group is studied.