Abstract
Let v be a valuation of a field K with a value group and a residue field , w be an extension of v to . Then w is called a residual algebraic torsion extension of v to if is an algebraic extension and is a torsion group. In this paper, a residual algebraic torsion extension of v to is described and its certain properties are investigated. Also, the existence of a residual algebraic torsion extension of a valuation on K to with given residue field and value group is studied. MSC:12J10, 12J20, 12F20.
Highlights
Let K be a field, v be a valuation on K with a value group Gv and a residue field kv
The big target is to define all extensions of v to K(x, . . . , xn)
A residual transcendental extension of v to K(x, . . . , xn) is defined in [ ] by Öke. These studies are summarized in the second section
Summary
Let K be a field, v be a valuation on K with a value group Gv and a residue field kv. Let I be a well-ordered set without the last element and (wi)i∈I be an ordered system of r.t. extensions of v to K(x), where wi is defined by a minimal pair (ai, δi) ∈ K × Gvfor all i ∈ I. Let um be an r.t. extension of v to K(xm) defined by a minimal pair (am, δm) ∈ K × Gvfor m = , .
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