Consider a field $k$ and its associated projective plane over $k$, $P^2(k)$.
Suppose we have the following properties $A1-A3$ of $P^2(k)$.
A1) There is a unique line $L(pq)$ through any two distinct points $p,q$ in $P^2(k)$.
A2) Every pair of two distinct lines intersects at exactly one point.
A3) Given any two triples collinear points $(p,q,r),(p',q',r')$, then $(L(pq')\cap L(p'q),L(pr')\cap L(rp'), L(qr')\cap L(rq'))$ are also colinear.(Pappus Hexagon Theorem)
Consider any 4 points in general position. The statement is that one can recover the field $k$ through those 4 points.
Q: What is the meaning of recovering here? Additive and multiplicative structure can be recovered?
Q': How to recover? Somehow this is obvious? This is not as obvious as elliptic curve to define group structure.
There is a method of coordinatizing a projective plane. This will give you an algebraic structure known as a planar ternary ring. Certain geometric properties of the plane are reflected in the algebraic structure or the ternary ring, for example a plane can be coordinatized by a skew field if and only if Desargues' theorem holds, and Pappus' theorem implies (in addition) that the plane can be coordinatized by a field.
The big issue is that you can have nonisomorphic planar ternary rings associated with the same plane, you are only guaranteed uniqueness up to isotopy. So the theorem of Pappus guarantees that coordinatizing will give you something isotopic to a field, and then I believe it is simple to show that you cannot have two nonisomorphic fields in the same isotopism class so the field is determined.
There are lots of references on coordinatizing projective planes, I would look these up on Google Scholar or something similar if you want more technical details.