valuation
Let $K$ be a field. A valuation^{} or absolute value^{} on $K$ is a function $\cdot :K\to \mathbb{R}$ satisfying the properties:

1.
$x\ge 0$ for all $x\in K$, with equality if and only if $x=0$

2.
$xy=x\cdot y$ for all $x,y\in K$

3.
$x+y\le x+y$
If a valuation satisfies $x+y\le \mathrm{max}(x,y)$, then we say that it is a nonarchimedean valuation. Otherwise we say that it is an archimedean valuation.
Every valuation on $K$ defines a metric on $K$, given by $d(x,y):=xy$. This metric is an ultrametric if and only if the valuation is nonarchimedean. 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 nonarchimedean 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 $\U0001d52d\subset K$ be a nonzero prime ideal^{1}^{1}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^{}., considered as a fractional ideal. For every nonzero element $x\in K$, let $r$ be the unique integer such that $x\in {\U0001d52d}^{r}$ but $x\notin {\U0001d52d}^{r+1}$. Define
$${x}_{\U0001d52d}:=\{\begin{array}{cc}1/N{(\U0001d52d)}^{r}\hfill & x\ne 0,\hfill \\ 0\hfill & x=0,\hfill \end{array}$$ 
where $N(\U0001d52d)$ denotes the absolute norm of $\U0001d52d$. Then $\cdot {}_{\U0001d52d}$ is a non–archimedean valuation on $K$, and furthermore every nonarchimedean valuation on $K$ is equivalent to $\cdot {}_{\U0001d52d}$ for some prime ideal $\U0001d52d$. 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 $\u2102$ yields a valuation of $K$ by way of the standard absolute value on $\mathbb{R}$ or $\u2102$, 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 $\u2102$. Such a prime is called real or complex according to whether the valuations comprising it arise from real or complex embeddings.
Title  valuation 
Canonical name  Valuation 
Date of creation  20130322 12:35:07 
Last modified on  20130322 12:35:07 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  17 
Author  djao (24) 
Entry type  Definition 
Classification  msc 13F30 
Classification  msc 13A18 
Classification  msc 12J20 
Classification  msc 11R99 
Synonym  absolute value 
Related topic  DiscreteValuationRing 
Related topic  DiscreteValuation 
Related topic  Ultrametric 
Related topic  HenselianField 
Defines  infinite prime 
Defines  finite prime 
Defines  archimedean 
Defines  nonarchimedean 
Defines  real prime 
Defines  complex prime 
Defines  prime 