## You are here

Homeplace as extension of homomorphism

## Primary tabs

# 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

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

of the field $k$ such that

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

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

# References

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

Related:

RamificationOfArchimedeanPlaces

Synonym:

extension theorem

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

13A18*no label found*12E99

*no label found*13F30

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections