Building a field given a geometry

Question

I took a course some years ago and in it was a treatment of how to associate a field to an abstract geometry.

I would very much appreciate some reading on this, as I have been unsuccessful on where to find any resources on such ideas, and I have since lost my notes!

Is there a book which treats, thoroughly, the connection between an abstract geometry and a field?

In particular:

Is there a way to consider a discrete geometry, like projective Steiner triple systems, and associate to these objects a field?

Any information would be extremely useful, introductory or extremely advanced. I would be particularly interested in anyones knowledge of resources which have connections to class field theory.

Thank you for your time and consideration.

Answer 1

For the most part, you can start with a geometric object such as a projective plane, and coordinatize. This leads to a structure called a planar ternary ring. The ring you get is not unique up to isomorphism, but rather up to a relation known as isotopism.

I'm not sure about other objects, you can also do this with biplanes though at least.

I don't have references on hand, but suggest searching for "planar ternary ring" along with the geometric structure you are interested in.

Answer 2

You may want to look at articles on Veblen-Wedderburn systems, Hall planes and Moufang planes to see if that helps your research. The book by Dembowski on Finite Geometries may interest you.