Let $K$ be a local field and $F/K$ be a cyclic extension of degree $n$, meaning that $n = [F:K]$ and $\operatorname{Gal}(F/K) \simeq C_n$ is cyclic.
In the proof of Lemma 2 of the paper "Euler Factors determine local Weil Representation" by Tim and Vladimir Dokchitser we considered a primitve character $\chi$ of $\operatorname{Gal}(F/K)$.
Question: What exactly is a primitive character? (Unfortunately, I was not able to find a proper definition somewhere.)
- Does it has something to do with the Galois group being cyclic? Or is there a more general definition?
- Do we need a Weil representation here (as this paper deals with them)?
- Are these related with Dirichlet characters? (My professor mentioned them once but I still have no idea what they are.)
It would be really nice if you could respond to my questions. Thank you!