On the validity of a Benz condition for non-standard Galois fields and for 2e

W. Benz showed that if a field K satisfies certain conditions, then every injection K² K² preserving Minkowsky distance 1 is semilinear up to translation. In this paper we consider one of these conditions. We prove that a non-standard Galois field associated with a Cauchy ultra filter over the set of all prime powers satisfies Benz condition but that the field of rationals does not. The last statement is proved by studying the set of rational points of a certain quartic curve, by considering a Weierstrass cubic birationally equivalent to that quartic.Dedicated to Professor M. Scafati Tallini on the occasion of her 65^th birthday