View source for Dedekind-theorem(2)

''on linear independence of field homomorphisms, Dedekind lemma''

Any set of field homomorphisms of a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200701.png" /> into another field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200702.png" /> is linearly independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200703.png" /> (see also [[Homomorphism|Homomorphism]]; [[Linear independence|Linear independence]]). I.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200704.png" /> are distinct homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200705.png" />, then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200706.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200707.png" />, not all zero, there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200708.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d1200709.png" /></td> </tr></table>

An immediate consequence is a basic estimate in [[Galois theory|Galois theory]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007011.png" /> are field extensions of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007012.png" /> and the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007014.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007015.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007016.png" /> (cf. [[Extension of a field|Extension of a field]]), than there are at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007017.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007018.png" />-homomorphisms of fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120070/d12007019.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Algebra" , '''2''' , Wiley  (1989)  pp. 81  (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.-H. Sprindler,   "Abstract algebra with applications" , '''2''' , M. Dekker  (1994)  pp. 395</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Jacobson,   "Lectures in abstract algebra: Theory of fields and Galois theory" , '''3''' , v. Nostrand  (1964)  pp. Chap. I, §3</TD></TR></table>