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[parent] place of field (Theorem)

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.

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 1   Let $\mathfrak{o}$ be a valuation domain with field of fractions $k$ and $\mathfrak{p}$ the maximal ideal of $\mathfrak{o}$ consisting of the non-units 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 residue-class 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 $\varphi(x^{-1}) = 0$

Bibliography

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




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See Also: Krull valuation, valuation determined by valuation domain, integrity characterized by places, ramification of archimedean places

Other names:  place, spot of field
Also defines:  place of field

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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, proof, maximal ideal, field of fractions, conversely, valuation domain, easy to see, ring homomorphism, restriction, subring, preimage, mapping, field
There are 101 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 7571 times total.

Classification:
AMS MSC12E99 (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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