The Beckman-Quarles theorem for mappings from ℝ2 to mathbbF2 {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

摘要

Let F be a subfield of a commutative field extending ℝ. Let We say thatf : 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 : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.