# Collineations of projective spaces and isomorphisms of fields

For a (topological) field $$F$$ by $$FP^2$$ we denote the projective plane, i.e., the quotient space of $$F^3\setminus\{0\}^3$$ by the equivalence relation $$\vec x\sim\vec y$$ iff $$\vec x=\lambda\vec y$$ for some $$\lambda\in F\setminus\{0\}$$.

A line in $$FP^2$$ is the image of $$L\setminus\{0\}^3$$ for some 2-dimensional linear subspace $$L$$ of $$F^3$$.

I need a reference to the following results, which (I hope) are true and known:

Theorem 1. Two fields $$F_1$$ and $$F_2$$ are isomorphic if and only if there exists a line-preserving bijection between the projective planes $$F_1P^2$$ and $$F_2P^2$$.

Theorem 2. Two topological fields $$F_1$$ and $$F_2$$ are topologically isomorphic if and only if there exists a line-preserving homeomorphism between the projective planes $$F_1P^2$$ and $$F_2P^2$$.

Remark. For finite fields, Theorem 1 holds in a stronger form: two finite fields $$F_1$$, $$F_2$$ are isomorphic if and only if there exists a bijection between the projective planes $$F_1P^2$$ and $$F_2P^2$$.

• The tag (geometry) is deprecated on MO, see the tag-info. Probably it could be replaced by some other suitable tag. (Or omitted entirely - as there already is a tag for algebraic geometry.) – Martin Sleziak Aug 22 at 7:43
• @MartinSleziak I noticed that the tag "geometry" is deprecated, but the system did not guggest anything more appropriate (now searching through all tags I found the appropriate tag "projective-geometry"). – Taras Banakh Aug 22 at 12:26