I'm trying to understand by myself Hopf-Galois theory, and for an example to verify certain properties I need a map to be a bijection. The details of my example are not important, what I just need is to know if the following statement is true or not (or in what conditions would it be true). The statement is:
Let $F,K$ be fields, with $K\subset F$. Let $a,b,c\in F$ such that $a\neq b$ and $c\neq 0$. Then $a\otimes_K c\neq b\otimes_K c$.
I feel this has to be true for my example to fit, but the tensor product makes me doubt if maybe $a\otimes c$ and $b\otimes c$ may not be different in general. I guess $F$ and $K$ being fields may be something that makes this statement true. I tried looking for a counterexample but had no luck.
In fact, in my example what I need is a bit more difficult, since I would need these things to be different (considering $a\neq b$): $$\sum_{\tau\in G}\tau(a)\otimes\tau(c)\neq \sum_{\tau\in G}\tau(b)\otimes \tau(c),$$ where $G=\text{Gal(F/K)}$ and $c$ is such that $\{\tau(c)\ |\ \tau\in G\}$ is a $K$-basis of $F$. I'm not sure if the statement I'm asking for will be enough for this to be true also, but in any case it would be a good starting point to continue thinking about my problem.
Is this true? How can I prove it if so? Any help or hint will be appreciated, thanks in advance.