The Beckman-quarles Theorem for Mappings from C to C

Let φ : C2×C2 → C, φ((x1, x2), (y1, y2)) = (x1−y1)2+(x2−y2)2. We say that f : C2 → C2 preserves distance d ≥ 0, if for each X, Y ∈ C2 φ(X, Y ) = d2 implies φ(f(X), f(Y )) = d2. We prove that each unit-distance preserving mapping f : C2 → C2 has a form I◦(γ, γ), where γ : C→ C is a field homomorphism and I : C2 → C2 is an affine mapping with orthogonal linear part. We prove an analogous result for mappings from K 2 to K , where K is a field such that char(K ) 6∈ {2, 3, 5} and −1 is a square. The classical Beckman-Quarles theorem states that each unit-distance preserving mapping from R to R (n ≥ 2) is an isometry, see [1]–[5]. Let φ : C × C → C, φ((x1, x2), (y1, y2)) = (x1 − y1) + (x2 − y2). We say that f : C → C preserves distance d ≥ 0, if for each X, Y ∈ C φ(X, Y ) = d implies φ(f(X), f(Y )) = d. If f : C → C and for each X, Y ∈ C φ(X,Y ) = φ(f(X), f(Y )), then f is an affine mapping with orthogonal linear part; it follows from a general theorem proved in [3, 58 ff], see also [4, p. 30]. The author proved in [9]: each unit-distance preserving mapping f : C → C satisfies (1) φ(X, Y ) = φ(f(X), f(Y )) for all X, Y ∈ C with rational φ(X,Y ). Theorem 1. If f : C → C preserves unit distance, f((0, 0)) = (0, 0), f((1, 0)) = (1, 0) and f((0, 1)) = (0, 1), then there exists a field homomorphism ρ : R → C satisfying ∀x1, x2 ∈ C: (2) f((x1, x2)) ∈ {(ρ(Re(x1)) + ρ(Im(x1)) · i, ρ(Re(x2)) + ρ(Im(x2)) · i), (ρ(Re(x1))− ρ(Im(x1)) · i, ρ(Re(x2))− ρ(Im(x2)) · i)}. Proof. Obviously, g = f|R2 : R → C preserves unit distance. The author proved in [8] that such a g has a form I ◦ (ρ, ρ), where ρ : R→ C is a field homomorphism and I : C → C is an affine mapping with orthogonal linear part. Since f((0, 0)) = (0, 0), f((1, 0)) = (1, 0), f((0, 1)) = (0, 1), we conclude that f|R2 = (ρ, ρ). From this, condition (2) holds true if (x1, x2) ∈ R. Assume now that (x1, x2) ∈ C \R2. 2000 Mathematics Subject Classification. 39B32, 51B20, 51M05.