My question was too big to put it in the title:
Having a set of linearly independent field monomorphisms $K→L$, is it still possible to express one of these monomorphisms as a linear combination of the others, only that the coefficients could vary depending on which element of $K$ we pick?
Say we have a set of $n$ independent field monomorphisms $K→L$, that is, the only way for $$c_1m_1(x)+c_2m_2(x)+...+c_nm_n(x)=0 $$ for all $x$ in $K$ is for all the $c_i$ to be zero.
But I was wondering, in contrast to vectors, is it still possible to express one of these monomorphisms, say $m_1(x)$, as a linear combination of the others?
To better explain myself, wouldn't it be possible for some fields $K$ to be divisible into two sets of elements, say $A$ and $B$, and then
$$m_1(a)=r_2m_2(a)+...+r_nm_n(a)$$ for all $a$ in $A$, and
$$m_1(b)=s_2m_2(b)+...+s_nm_n(b)$$ for all $b$ in $B$.
Notice that this doesn't contradict (at least for what I can see), the fact that the set of monomorphisms is linearly independent.
I would really appreciate any help/thoughts