# place as extension of homomorphism

###### Theorem.

If $f$ is a ring homomorphism^{} from a subring $\U0001d52c$ 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

$${\phi |}_{\U0001d52c}=f.$$ |

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 |
---|---|

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 |