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# 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*

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new correction: examples and OEIS sequences by fizzie

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new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith