The Beckman-Quarles theorem for mappings from ℝ² to${\mathbb{F}}^2$, where F is a subfield of a commutative field extending ℝ, where F is a subfield of a commutative field extending ℝ

Abstract

Let F be a subfield of a commutative field extending ℝ. Let$\varphi _2 :{\mathbb{F}}^2 x {\mathbb{F}}^2 \to {\mathbb{F}},\varphi _2 ((x_1 ,x_2 ), (y_1 ,y_2 )) = (x_1 - y_1 )^2 + (x_2 - y_2 )^2$ We say thatf :

${\mathbb{R}}^2 \to {\mathbb{F}}^2$ preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ₂(f(x),f(y)) = d² . We prove that each unit-distance preserving mappingf :${\mathbb{R}}^2 \to {\mathbb{F}}^2$ has a formI o (ρ,ρ), where$\rho :{\mathbb{R}} \to {\mathbb{F}}$ is a field homomorphism and$I:{\mathbb{F}}^2 \to {\mathbb{F}}^2$ is an affine mapping with orthogonal linear part.