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valuation (Definition)

Let $ K$ be a field. A valuation on $ K$ is a function $ \vert\cdot\vert\colon K \to \mathbb{R}$ satisfying the properties:

  1. $ \vert x\vert \geq 0$ for all $ x \in K$, with equality if and only if $ x=0$
  2. $ \vert xy\vert = \vert x\vert\cdot \vert y\vert$ for all $ x,y \in K$
  3. $ \vert x+y\vert \leq \vert x\vert + \vert y\vert$
If a valuation satisfies $ \vert x+y\vert \leq \max(\vert x\vert, \vert y\vert)$, then we say that it is a non-archimedean valuation. Otherwise we say that it is an archimedean valuation.

Every valuation on $ K$ defines a metric on $ K$, given by $ d(x,y) := \vert x-y\vert$. This metric is an ultrametric if and only if the valuation is non-archimedean. Two valuations are equivalent if their corresponding metrics induce the same topology on $ K$. An equivalence class $ v$ of valuations on $ K$ is called a prime of $ K$. If $ v$ consists of archimedean valuations, we say that $ v$ is an infinite prime, or archimedean prime. Otherwise, we say that $ v$ is a finite prime, or non-archimedean prime.

In the case where $ K$ is a number field, primes as defined above generalize the notion of prime ideals in the following way. Let $ \mathfrak{p}\subset K$ be a nonzero prime ideal 1, considered as a fractional ideal. For every nonzero element $ x \in K$, let $ r$ be the unique integer such that $ x \in \mathfrak{p}^r$ but $ x \notin \mathfrak{p}^{r+1}$. Define

\begin{displaymath} \vert x\vert _\mathfrak{p}:= \begin{cases} 1/N(\mathfrak{p})^r & x \neq 0, \ 0 & x=0, \end{cases}\end{displaymath}
where $ N(\mathfrak{p})$ denotes the absolute norm of $ \mathfrak{p}$. Then $ \vert\cdot\vert _\mathfrak{p}$ is a non-archimedean valuation on $ K$, and furthermore every non-archimedean valuation on $ K$ is equivalent to $ \vert\cdot\vert _\mathfrak{p}$ for some prime ideal $ \mathfrak{p}$. Hence, the prime ideals of $ K$ correspond bijectively with the finite primes of $ K$, and it is in this sense that the notion of primes as valuations generalizes that of a prime ideal.

As for the archimedean valuations, when $ K$ is a number field every embedding of $ K$ into $ \mathbb{R}$ or $ \mathbb{C}$ yields a valuation of $ K$ by way of the standard absolute value on $ \mathbb{R}$ or $ \mathbb{C}$, and one can show that every archimedean valuation of $ K$ is equivalent to one arising in this way. Thus the infinite primes of $ K$ correspond to embeddings of $ K$ into $ \mathbb{R}$ or $ \mathbb{C}$, and we call such a prime real or complex according to whether the valuations comprising it arise from real or complex embeddings.


Footnotes

... ideal1
By “prime ideal” we mean “prime fractional ideal of $ K$” or equivalently “prime ideal of the ring of integers of $ K$”. We do not mean literally a prime ideal of the ring $ K$, which would be the zero ideal.


"valuation" is owned by djao.
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See Also: discrete valuation ring, discrete valuation, ultrametric, henselian field

Also defines:  infinite prime, finite prime, archimedean, non-archimedean, real prime, complex prime, prime

Attachments:
independence of valuations (Theorem) by pahio
Gelfand-Tornheim theorem (Theorem) by pahio
trivial valuation (Definition) by pahio
equivalent valuations (Definition) by pahio
Krull valuation (Definition) by pahio
alternative definition of valuation (Definition) by rspuzio
alternative definition of Krull valuation (Definition) by polarbear
properties of non-archimedean valuations (Theorem) by rm50
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Cross-references: complex embeddings, complex, real, absolute value, embedding, absolute norm, integer, fractional ideal, zero ideal, ring, mean, prime ideals, number field, equivalence class, topology, induce, equivalent, ultrametric, metric, equality, properties, function, field
There are 128 references to this entry.

This is version 12 of valuation, born on 2002-04-15, modified 2006-04-11.
Object id is 2835, canonical name is Valuation.
Accessed 22668 times total.

Classification:
AMS MSC: 11R99 (Number theory :: Algebraic number theory: global fields :: Miscellaneous)
 12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
 13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)
 13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)
Pending Errata and Addenda
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