A short proof for the relation between Weil indices and factors

Let F be a finite extension of ℚ_p and let ψ be a non-trivial character of F. For a∈F* let γ(a,ψ) be the normalized Weil index splitting the Hilbert symbol. In this short note we give a simple proof for the relation (Formula presented.) where η_a is the quadratic character of F* whose kernel is N(F√a) and where &(⋅,⋅,⋅) is the epsilon factor appearing in Tate’s thesis.