Valuation (algebra)

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. In all of these examples, the valuation assumes integer values and is therefore called a discrete valuation, but in general, the integers are replaced by an abelian totally ordered group.

A field with a valuation on it is called a valued field.

Definition

Discrete valuations

A discrete valuation on a field K is a function:

ν

:

K

→

Z

∪

{

∞

}

{\displaystyle \nu :K\to \mathbb {Z} \cup \{\infty \}}

satisfying the conditions:

ν

(

x

⋅

y

)

=

ν

(

x

)

+

ν

(

y

)

{\displaystyle \nu (x\cdot y)=\nu (x)+\nu (y)}

ν

(

x

+

y

)

≥

min

{

ν

(

x

)

,

ν

(

y

)

}

{\displaystyle \nu (x+y)\geq \min {\big \{}\nu (x),\nu (y){\big \}}}

ν

(

x

)

=

∞

⟺

x

=

0

{\displaystyle \nu (x)=\infty \iff x=0}

for all

x

,

y

∈

K

{\displaystyle x,y\in K}

.

Note that often the trivial valuation which takes on only the values

0

,

∞

{\displaystyle 0,\infty }

is explicitly excluded.

A field with a non-trivial discrete valuation is called a discrete valuation field.

Relation to discrete valuation rings

To every field

K

{\displaystyle K}

with discrete valuation

ν

{\displaystyle \nu }

we can associate the subring

O

ν

:=

{

x

∈

K

∣

ν

(

x

)

≥

0

}

{\displaystyle {\mathcal {O}}_{\nu }:=\left\{x\in K\mid \nu (x)\geq 0\right\}}

of

K

{\displaystyle K}

, which is a discrete valuation ring. Conversely, the valuation

ν

:

A

→

Z

∪

{

∞

}

{\displaystyle \nu :A\rightarrow \mathbb {Z} \cup \{\infty \}}

on a discrete valuation ring

A

{\displaystyle A}

can be extended in a unique way to a discrete valuation on the quotient field

K

=

Quot

(

A

)

{\displaystyle K={\text{Quot}}(A)}

; the associated discrete valuation ring

O

ν

{\displaystyle {\mathcal {O}}_{\nu }}

is just

A

{\displaystyle A}

.

Discrete valuation rings

O

ν

{\displaystyle {\mathcal {O}}_{\nu }}

are local rings with maximal ideal

m

ν

:=

{

x

∈

O

ν

∣

ν

(

x

)

>

0

}

,

{\displaystyle {\mathfrak {m}}_{\nu }:=\left\{x\in {\mathcal {O}}_{\nu }\mid \nu (x)>0\right\},}

so there is a notion of residue field

κ

=

O

ν

/

m

ν

{\displaystyle \kappa ={\mathcal {O}}_{\nu }/{\mathfrak {m}}_{\nu }}

.