The Beckman-Quarles theorem for mappings from R² to F², where F is a subfield of a commutative field extending R

الأصل · EN

Let F be a subfield of a commutative field extending R. Let ϕ₂: F² × F² → F, ϕ₂((x₁,x₂),(y₁,y₂))=(x₁-y₁)²+(x₂-y₂)². We say that f:R² → F² preserves distance d ≥ 0 if for each x,y ∈ R² |x-y|=d implies ϕ₂(f(x),f(y))=d². We prove that each unit-distance preserving mapping f:R² → F² has a form I ∘ (ρ,ρ), where ρ: R → F is a field homomorphism and I: F² → F² is an affine mapping with orthogonal linear part.