In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function
ν
:
K
→
Z
∪
{
∞
}
satisfying the conditions
ν
(
x
⋅
y
)
=
ν
(
x
)
+
ν
(
y
)
ν
(
x
+
y
)
≥
min
{
ν
(
x
)
,
ν
(
y
)
}
ν
(
x
)
=
∞
⟺
x
=
0
for all
x
,
y
∈
K
.
Note that often the trivial valuation which takes on only the values
0
,
∞
is explicitly excluded.
A field with a non-trivial discrete valuation is called a discrete valuation field.
Discrete valuation rings and valuations on fields
To every field
K
with discrete valuation
ν
we can associate the subring
of
K
, which is a discrete valuation ring. Conversely, the valuation
ν
:
A
→
Z
∪
{
∞
}
on a discrete valuation ring
A
can be extended in a unique way to a discrete valuation on the quotient field
K
=
Quot
(
A
)
; the associated discrete valuation ring
O
K
is just
A
.
For a fixed prime
p
and for any element
x
∈
Q
different from zero write
x
=
p
j
a
b
with
j
,
a
,
b
∈
Z
such that
p
does not divide
a
,
b
, then
ν
(
x
)
=
j
is a discrete valuation on
Q
, called the p-adic valuation.
Given a Riemann surface
X
, we can consider the field
K
=
M
(
X
)
of meromorphic functions
X
→
C
∪
{
∞
}
. For a fixed point
p
∈
X
, we define a discrete valuation on
K
as follows:
ν
(
f
)
=
j
if and only if
j
is the largest integer such that the function
f
(
z
)
/
(
z
−
p
)
j
can be extended to a holomorphic function at
p
. This means: if
ν
(
f
)
=
j
>
0
then
f
has a root of order
j
at the point
p
; if
ν
(
f
)
=
j
<
0
then
f
has a pole of order
−
j
at
p
. In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point
p
on the curve.
More examples can be found in the article on discrete valuation rings.
