I first remind Dedekind's independence theorem
Let $G$ be a group and $\sigma_1,\cdots,\sigma_n$ characters of $G$ to a field $L$ that are $2$ by $2$ different. Then $\{\sigma_1,\cdots,\sigma_n\}$ is a linearily independent set over $L$.
Where
A character $\sigma$ of $G$ to $L$ is a group morphism $\sigma$ from $G$ to the multiplicative group $L^*$.
and
A set of characters $\sigma_1,\cdots,\sigma_n$ of $G$ to $L$ is called linearilly independent if $$a_1\sigma_1 + \cdots a_n\sigma_n = 0$$
implies that $a_1 = \cdots = a_n = 0$
Apparently a consequence of this is the fact that distinct ring homomorphisms of a field $K$ into a field $F$ are linearily independent over $F$.
I don't get this.
Indeed, ring homomorphisms are not from $(K^*,\cdot)$ to $(F^*,\cdot)$ so we don't fit into the setting of Dedekind's theorem.