Local Artin root numbers associated to some classical polynomials (Q916709)

The paper studies Hilbert polynomials F(T,X)\(\in {\mathbb{Q}}(T)[X]\) having Galois group over \({\mathbb{Q}}(T)\) isomorphic to the alternating group \(A_ n\). Furthermore, the exponential Taylor polynomials f(X) considered by Schur with Galois group of f(X) over \({\mathbb{Q}}\) isomorphic to \(A_ n\), respectively the symmetric group \(S_ n\), are investigated. For each one of these polynomials the local Artin root numbers of the associated Galois representation are determined. Moreover, Weil's additive characters of the respective Witt classes are computed.