Background
Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like "intuitively..." or "morally...". These answers often can provide useful heuristics/analogies/intuition for reasoning about mathematics. Such questions I believe prompt useful discussion and can generate insights, and this is the sort of answer I'd like here.
Question(s)
Define a character $\chi$ of a group with values in the field $L$ to be a homomorphism
$$\chi\colon G \longrightarrow L^{\times}$$
We call the collection $\chi_1, \chi_2, \dots, \chi_n$ linearly independent (as functions) over $L$ if
$$a_1\chi_1 + a_2\chi_2 + \dots + a_n\chi_n = 0$$ for $a_i \in L$ implies that $a_1 = \dots = a_n = 0$.
There is a result that states that if $\chi_1, \chi_2, \dots, \chi_n$ are distinct characters of $G$ with values in $L$ then they are linearly independent over $L$. Of course there are formal proofs for this fact, one of which can be found in Dummit and Foote, Sec 14.2. However I feel that, given the amount of structure on the algebraic objects in this picture ($L$ a field, $G$ a group) there is a "reason" why this is true.
This result is important because it is used in the proof of the following theorem (which is a precursor to the fundamental theorem of Galois Theory):
Theorem: Let $G = \{\sigma_1 = \mathrm{id}, \sigma_2, \dots,\sigma_n\}$ be a subgroup of automorphisms of a field $K$ and let $F$ be the fixed field. Then $[K\colon F] = n = |G|$.
To prove this theorem, Dummit and Foote use the fact that distinct embeddings $\sigma_1, \dots, \sigma_n$ of a field $K$ into a field $L$ are linearly independent as functions on $K$. In particular, distinct automorphisms of $K$ have this property.
The main thing:
It seems there is some "thing" about the structure of fields that prevents characters and field isomorphisms/embeddings from being linear combinations of one another. Why is this so? Is this the case with weaker structures than a field as the image set? Can I represent field isomorphisms as combinations of higher degrees of one another? I.e is it possible for there to be some polynomial $f(x_1, \dots, x_n) \in F[x_1, \dots, x_n]$ such that
$$f(\chi_1, \chi_2, \dots, \chi_n) = 0$$
In summary, it seems that the structures surrounding Galois theory are somehow inter-related in some way I am yet to perceive, and I would like more insight on this matter.