You are here
Homeplace of field
Primary tabs
place of field
Let $F$ be a field and $\infty$ an element not belonging to $F$. The mapping
$\varphi:\,k\to F\cup\{\infty\},$ 
where $k$ is a field, is called a place of the field $k$, if it satisfies the following conditions.

The restriction $\varphi_{\mathfrak{o}}$ is a ring homomorphism from $\mathfrak{o}$ to $F$.

If $\varphi(a)=\infty$, then $\varphi(a^{{1}})=0$.
It is easy to see that the subring $\mathfrak{o}$ of the field $k$ is a valuation domain; so any place of a field determines a unique valuation domain in the field. Conversely, every valuation domain $\mathfrak{o}$ with field of fractions $k$ determines a place of $k$:
Theorem.
Let $\mathfrak{o}$ be a valuation domain with field of fractions $k$ and $\mathfrak{p}$ the maximal ideal of $\mathfrak{o}$, consisting of the nonunits of $\mathfrak{o}$. Then the mapping
$\varphi:\,k\to\mathfrak{o/p}\cup\{\infty\}$ 
defined by
$\varphi(x):=\begin{cases}x+\mathfrak{p}\quad\mathrm{when}\,\,\,x\in\mathfrak{o% },\\ \infty\quad\mathrm{when}\,\,\,x\in k\smallsetminus\mathfrak{o},\end{cases}$ 
is a place of the field $k$.
Proof. Apparently, $\varphi^{{1}}(\mathfrak{o/p})=\mathfrak{o}$ and the restriction $\varphi_{\mathfrak{o}}$ is the canonical homomorphism from the ring $\mathfrak{o}$ onto the residueclass ring $\mathfrak{o/p}$. Moreover, if $\varphi(x)=\infty$, then $x$ does not belong to the valuation domain $\mathfrak{o}$ and thus the inverse element $x^{{1}}$ must belong to it without being its unit. Hence $x^{{1}}$ belongs to the ideal $\mathfrak{p}$ which is the kernel of the homomorphism $\varphi\mathfrak{o}$. So we see that $\varphi(x^{{1}})=0$.
References
 1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
Mathematics Subject Classification
13F30 no label found13A18 no label found12E99 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Prime numbers out of sequence by Rubens373
Oct 7
new question: Lorenz system by David Bankom
Oct 19
new correction: examples and OEIS sequences by fizzie
Oct 13
new correction: Define Galois correspondence by porton
Oct 7
new correction: Closure properties on languages: DCFL not closed under reversal by babou
new correction: DCFLs are not closed under reversal by petey
Oct 2
new correction: Many corrections by Smarandache
Sep 28
new question: how to contest an entry? by zorba
new question: simple question by parag