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
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