Difference between revisions of "Dedekind-theorem(2)"
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''on linear independence of field homomorphisms, Dedekind lemma'' | ''on linear independence of field homomorphisms, Dedekind lemma'' | ||
| − | Any set of field homomorphisms of a [[Field|field]] | + | Any set of field homomorphisms of a [[Field|field]] $E$ into another field $F$ is linearly independent over $F$ (see also [[Homomorphism|Homomorphism]]; [[Linear independence|Linear independence]]). I.e., if $\sigma _ { 1 } , \ldots , \sigma _ { t }$ are distinct homomorphisms $E \rightarrow F$, then for all $a _ { 1 } , \dots , a _ { t }$ in $F$, not all zero, there is an $u \in E$ such that |
| − | + | \begin{equation*} a _ { 1 } \sigma _ { 1 } ( u ) + \ldots + a _ { t } \sigma _ { t } ( u ) \neq 0. \end{equation*} | |
| − | An immediate consequence is a basic estimate in [[Galois theory|Galois theory]]: If | + | An immediate consequence is a basic estimate in [[Galois theory|Galois theory]]: If $E$, $F$ are field extensions of a field $K$ and the degree $[ E : K ]$ of $E$ over $K$ is $n$ (cf. [[Extension of a field|Extension of a field]]), than there are at most $n$ $K$-homomorphisms of fields $E \rightarrow F$. |
====References==== | ====References==== | ||
| − | <table>< | + | <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> |
Latest revision as of 16:55, 1 July 2020
on linear independence of field homomorphisms, Dedekind lemma
Any set of field homomorphisms of a field $E$ into another field $F$ is linearly independent over $F$ (see also Homomorphism; Linear independence). I.e., if $\sigma _ { 1 } , \ldots , \sigma _ { t }$ are distinct homomorphisms $E \rightarrow F$, then for all $a _ { 1 } , \dots , a _ { t }$ in $F$, not all zero, there is an $u \in E$ such that
\begin{equation*} a _ { 1 } \sigma _ { 1 } ( u ) + \ldots + a _ { t } \sigma _ { t } ( u ) \neq 0. \end{equation*}
An immediate consequence is a basic estimate in Galois theory: If $E$, $F$ are field extensions of a field $K$ and the degree $[ E : K ]$ of $E$ over $K$ is $n$ (cf. Extension of a field), than there are at most $n$ $K$-homomorphisms of fields $E \rightarrow F$.
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