# Dedekind-theorem(2)

From Encyclopedia of Mathematics

*on linear independence of field homomorphisms, Dedekind lemma*

Any set of field homomorphisms of a field into another field is linearly independent over (see also Homomorphism; Linear independence). I.e., if are distinct homomorphisms , then for all in , not all zero, there is an such that

An immediate consequence is a basic estimate in Galois theory: If , are field extensions of a field and the degree of over is (cf. Extension of a field), than there are at most -homomorphisms of fields .

#### References

[a1] | P.M. Cohn, "Algebra" , 2 , Wiley (1989) pp. 81 (Edition: Second) |

[a2] | K.-H. Sprindler, "Abstract algebra with applications" , 2 , M. Dekker (1994) pp. 395 |

[a3] | N. Jacobson, "Lectures in abstract algebra: Theory of fields and Galois theory" , 3 , v. Nostrand (1964) pp. Chap. I, §3 |

**How to Cite This Entry:**

Dedekind-theorem(2).

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dedekind-theorem(2)&oldid=15461

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article