Dedekind-theorem(2)

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