Valuations in algebraic field extensions

• Published in: Journal of algebra Vol. 312; no. 2; pp. 1033 - 1074
• Main Authors: Herrera Govantes, F.J., Olalla Acosta, M.A., Spivakovsky, M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-06-2007; Elsevier
• Subjects: Algebraic extension; Algebraic Geometry; Commutative Algebra; Key polynomial; Mathematics; Newton polygon; Valuation; Newton polygon; Key polynomial; Valuation; Algebraic extension; Valuations
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2007.02.022

Let
K
→
L
be an algebraic field extension and
ν a valuation of
K. The purpose of this paper is to describe the totality of extensions
{
ν
′
}
of
ν to
L using a refined version of MacLane's key polynomials. In the basic case when
L is a finite separable extension and
rk
ν
=
1
, we give an explicit description of the limit key polynomials (which can be viewed as a generalization of the Artin–Schreier polynomials). We also give a realistic upper bound on the order type of the set of key polynomials. Namely, we show that if
char
K
=
0
then the set of key polynomials has order type at most
N
, while in the case
char
K
=
p
>
0
this order type is bounded above by
(
[
log
p
n
]
+
1
)
ω
, where
n
=
[
L
:
K
]
. Our results provide a new point of view of the well-known formula
∑
j
=
1
s
e
j
f
j
d
j
=
n
and the notion of defect.