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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 $F_2P^2$.

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  • $\begingroup$ 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.) $\endgroup$ – Martin Sleziak Aug 22 at 7:43
  • $\begingroup$ @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"). $\endgroup$ – Taras Banakh Aug 22 at 12:26
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Theorem 1 is due to Hilbert and proven in Hartshorne's book on projective planes. In fact, every choice of projective plane and 4 points in that plane, with no 3 in a line, gives an explicit construction of a "ternary ring" (a weakened notion of ring), and if the projective plane is defined over a field, that ring is the field.

If Theorem 2 is known to be true, I would guess you would find it in H. Salzmann et. al., Compact Projective Planes With an Introduction to Octonion Geometry, but I am travelling so I can't check.

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  • $\begingroup$ Thank you for the answer. Concerning Theorem 2 and the book of Salzmann, probably you are right. Unfortunately, google book does not show the necessary pages (212-232) on which there is a proof of topological version. So, I will wait for a more extensive answer. $\endgroup$ – Taras Banakh Aug 22 at 12:31

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