On the validity of a Benz condition for non-standard Galois fields and for \(\mathbb{Q}\) (Q1912006)

The following theorem is known [\textit{W. Benz}, J. Geom. 17, 193-201 (1981; Zbl 0499.51005) or his book: `Geometrische Transformationen' (1992; Zbl 0754.51005)]: In a field \(K\) with \(\text{char} K \neq 2,3,5\) and such that (i) --3 is not a square (ii) \(K = \{x + {4x \over (x - 1) (1 - y^2)}\): \(x,y \in K\), \(x \neq 1 \neq y^2\}\) every injection \(\sigma : K^2 \to K^2\) preserving Minkowski distance 1 is semilinear up to a translation. In this paper the condition (ii) is considered. It is proved that a nonstandard Galois field [see \textit{G. Tallini}, Conf. Semin. Mat. Univ. Bari 209, 17 p. (1986; Zbl 0663.12021)] associated with a Cauchy ultra filter over the set of all primitive powers satisfies the property (ii). On the other hand is shown that the field \(Q\) of rationals does not satisfy condition (ii). This is proved by studying the existence or non-existence of rational points of the Weierstrass cubic birationally equivalent to the quartic \((x - c) (x - 1) (1 - y^2) + 4x = 0\).