# place as extension of homomorphism

###### Theorem.

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

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

of the field $k$ such that

 $\varphi|_{\mathfrak{o}}=f.$

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

## References

• 1 Emil Artin: .  Lecture notes.  Mathematisches Institut, Göttingen (1959).
Title place as extension of homomorphism PlaceAsExtensionOfHomomorphism 2013-03-22 14:57:21 2013-03-22 14:57:21 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 13A18 msc 12E99 msc 13F30 extension theorem RamificationOfArchimedeanPlaces