A metric symbol for pairs of polynomials over local fields (Q2707864)

Let \(f\) and \(g\) be two polynomials of prime degree \(q\) over a local field \((K, v)\). The author defines a metric symbol \(({g\over f})\) which involves the discriminant of \(f\) and the resultant of \(f\) and \(g\). Although the definition is not symmetric in \(f\) and \(g\), the symbol enjoys some interesting properties: an irreducibility criterion, it is transitive and it satisfies a reciprocity law. As an application, it is proved that the quartic Diophantine equation NEWLINE\[NEWLINE(X ^2 + Y ^2 - Z ^2 - n) ^2 + 4(Y-Z) (Y Z ^2 - X^2 Y - Y ^2 Z + n Z) = 0NEWLINE\]NEWLINE satisfies the local-global principle.