Tits proved that (sufficiently high rank) spherical buildings arise from an algebraic group and a field, so any building is some $\Delta(G, F)$. He also showed that a building isomorphism $\Delta(G,F)\simeq\Delta(G',F')$ induces a field isomorphism $F\to F'$.


This shows that the field is somehow coded up in the isomorphism type of the building. I'm wondering whether a construction is floating around anywhere that shows how to construct the field from the combinatorics of the building.


To help clarify what I'm after: I've seen a construction of the real field from the incidence structure or geometry of the real projective plane (pick a pair of lines, prove they're bijective, define addition via some more lines and unique intersection points which must exist, etc.). In the end you have parametrized a projective line by matching up points on it with the underlying field. The construction requires a choice of a few points in general position and so is given sort of "up to collineation". I'm under the impression that I should view those results on spherical buildings as generalizations of results in projective geometry, and I'm looking for an analogous construction. That is, a construction of a field in terms of the apartments, relations between them, etc. In the end presumably some collection of objects is parametrized by the underlying field. Or is this not the case?



Constructing a field from a spherical building
The building in question must be of sufficiently high rank. Projective planes give rise to buildings (of type $A\_2$), but need not come from groups or fields. There is [plenty](http://en.wikipedia.org/wiki/Non-Desarguesian_plane) of examples of this sort.
Same applies to generalized quadrangles (buildings of type $C\_2$).
And there are also "trivial" rank 2 buildings of type $A\_1\times A\_1$, which need not come from fields.


Thus if you want to reconstruct a field from a rank 2 building, you will need an extra geometric structure; e.g. in the case of $A\_2$, you need e.g. a [Pappian axiom](http://en.wikipedia.org/wiki/Pappus%2527s_hexagon_theorem) to hold.  

It gets better with rank greater than 2. E.g. if you have an $A\_3$ as a sub-building, then one can use a standard projective geometry argument, which shows that one has a projective space over a division ring. (Yes, you must allow for such a possibility, commutativity would not come from building axioms alone).


