PlanetMath: place of field

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place of field

(Theorem)

Let be a field and $\infty$ an element not belonging to . The mapping

$\displaystyle \varphi: \,k\to F\cup\{\infty\},$

where

is a field, is called a place of the field

, if it satisfies the following conditions.

The preimage $\varphi^{-1}(F) = \mathfrak{o}$ is a subring of .
The restriction $\varphi\vert _\mathfrak{o}$ is a ring homomorphism from $\mathfrak{o}$ to .
If $\varphi(a) = \infty$ , then $\varphi(a^{-1}) = 0$ .

It is easy to see that the subring $\mathfrak{o}$ of the field 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 determines a place of :

Theorem 1 Let $\mathfrak{o}$ be a valuation domain with field of fractions

and $\mathfrak{p}$ the maximal ideal of $\mathfrak{o}$ , consisting of the non-units of $\mathfrak{o}$ . Then the mapping

$\displaystyle \varphi: \,k\to \mathfrak{o/p}\cup\{\infty\}$

defined by

$\begin{displaymath} \varphi(x):= \begin{cases} x+\mathfrak{p} \quad \mathrm{when... ...hrm{when} \,\,\, x \in k\smallsetminus\mathfrak{o}, \end{cases}\end{displaymath}$

is a place of the field

Proof. Apparently, $\varphi^{-1}(\mathfrak{o/p}) = \mathfrak{o}$ and the restriction $\varphi\vert _\mathfrak{o}$ is the canonical homomorphism from the ring $\mathfrak{o}$ onto the residue-class ring $\mathfrak{o/p}$ . Moreover, if $\varphi(x) = \infty$ , then 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 $\varphi\vert\mathfrak{o}$ . So we see that $\varphi(x^{-1}) = 0$ .

Bibliography

1: Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).

"place of field" is owned by pahio.

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Other names:

place, spot of field

Also defines:

place of field

This object's parent.

Attachments:: place as extension of homomorphism (Theorem) by pahio

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Cross-references: kernel, ideal, unit, inverse, residue-class ring, onto, ring, homomorphism, canonical, maximal ideal, field of fractions, valuation domain, easy to see, ring homomorphism, restriction, subring, preimage, mapping, field
There are 58 references to this entry.

This is version 13 of place of field, born on 2005-01-13, modified 2006-12-12.
Object id is 6640, canonical name is PlaceOfField.
Accessed 5401 times total.

Classification:

AMS MSC:	12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)
	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)

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