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:



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> **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 $F\_2P^2$.
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> **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 $F\_2P^2$.
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**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 $F\_2P^2$.*



Collineations of projective spaces and isomorphisms of fields
Theorem 1 is due to Hilbert and proven in Hartshorne's [book](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.475.3070&rep=rep1&type=pdf) 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.


