Throughout this paper, *v* is a valuation of a field *K* with a value group
, a valuation ring
and a residue field
,
is an algebraic closure of *K*,
is a fixed extension of *v* to
. The value group of
is the divisible closure of
and its residue field is the algebraic closure of
.
and
are rational function fields over *K* with one and *n* variables respectively. For any *α* in
,
denotes its natural image in
. If
, then the restriction of
to
will be denoted by
.

Let *w* be an extension of *v* to
. Then *w* is called a residual transcendental (r.t.) extension of *v* if
is a transcendental extension.

The valuation *w*, which is defined for each
as
is called Gauss extension of *v* to
, its residue field is
, is the simple transcendental extension of
and
[5].

The valuation

, which is defined for each

as

is called a valuation defined by the pair
or
is called a pair of definitions of *w*. Also, *w* is an r.t. extension of *v*. If
for every
such that
, then
is called a minimal pair with respect to *K*[2].

If

*w* is an r.t. extension of

*v* to

, there exists a minimal pair

such that

*a* is separable over

*K*. Two pairs

and

define the same valuation

*w* if and only if

and

[

2]. Let

be the minimal polynomial of

*a* with respect to

*K* and

. For each

, let

, where

,

,

, be the

*f*-expansion of

*F*. Then

*w* is defined as follows:

Then
. Let *e* be the smallest non-zero positive integer such that
. Then there exists
such that
,
and
is an element of
and
is transcendental over
.
can be identified canonically with the algebraic closure of
in
and
[2].

Let *w* be an extension of *v* to
. *w* is called a residual algebraic (r.a.) extension of *v* if
is an algebraic extension. If *w* is an r.a. extension of *v* to
and
is not a torsion group, then *w* is called a residual algebraic free (r.a.f.) extension of *v*. In this case, the quotient group
is a free abelian group. More precisely,
is isomorphic to *Z*[3].

*w* is called a residual algebraic torsion (r.a.t) extension of *v* if *w* is an r.a. extension of *v* and
is a torsion group. In this case,
is satisfied [3].

The order relation on the set of all r.t. extensions of *v* to
is defined as follows:
for all polynomials
. If
and there exists
such that
, then it is written
. Let
be minimal pairs of the definition of the r.t. extensions
and
of *v* to
, respectively. Then
if and only if
and
; moreover,
if and only if
and
[3].

Let *I* be a well-ordered set without the last element and
be an ordered system of r.t. extensions of *v* to
, where
is defined by a minimal pair
for all
. If
for all
, then
is called an ordered system of r.t. extensions of *v* to
.

Then the valuation of

defined as

for all
is an extension of *v* to
and it is called a limit of the ordered system
. *w* may not be an r.t. extension of *v* to
[3].

Using the above studies an r.a.t extension of *v* to
can be defined. For this reason the r.t. extension of *v* to
defined in [4] can be used. An r.t. extension of *v* to
is defined by using r.t. extensions of *v* to
for
in [4].

Let
be an r.t. extension of *v* to
defined by a minimal pair
for
, where
and
,
for
. Each polynomial
can be uniquely written as
, where
,
for
.

The valuation

*w* defined as

is an extension of *v* to
. *u* is an r.t. extension of *v* which is a common extension of
to
. Then
. Let
be the smallest positive integer such that
, where
is the restriction of
to
. Then there exists
such that
,
,
and
is transcendental over
for
.
can be canonically identified with the algebraic closure of
in
and
[4].

In the next section, an r.a.t extension of *v* to
will be defined by using that r.t. extension.