PlanetMath: place as extension of homomorphism

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place as extension of homomorphism

(Theorem)

Theorem 1 If

is a ring homomorphism from a subring $\mathfrak{o}$ of a field

such that

, then there exists a place

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

of the field

such that

$\displaystyle \varphi\vert _\mathfrak{o} = f.$

Note. That should be algebraically closed, does not mean any restriction, since every field is extendable to such one.

Bibliography

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

"place as extension of homomorphism" is owned by pahio.

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Other names:

extension theorem

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Cross-references: algebraically closed, field, subring, ring homomorphism
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This is version 6 of place as extension of homomorphism, born on 2005-01-19, modified 2005-03-17.
Object id is 6651, canonical name is PlaceAsExtensionOfHomomorphism.
Accessed 1714 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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