place as extension of homomorphism

Theorem.

If $f$ is a ring homomorphism from a subring $𝔬$ of a field $k$ to an algebraically closed field $F$ such that  $f(1)=1$,  then there exists a place (http://planetmath.org/PlaceOfField)

$$\phi :k\to F\cup \{\mathrm{\infty }\}$$

of the field $k$ such that

$${\phi |}_{𝔬}=f.$$

Note.  That $F$ should be algebraically closed, does not , since every field is extendable to such one.

Title	place as extension of homomorphism
Canonical name	PlaceAsExtensionOfHomomorphism
Date of creation	2013-03-22 14:57:21
Last modified on	2013-03-22 14:57:21
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	9
Author	pahio (2872)
Entry type	Theorem
Classification	msc 13A18
Classification	msc 12E99
Classification	msc 13F30
Synonym	extension theorem
Related topic	RamificationOfArchimedeanPlaces