My question is a doubt I had in the last point to the first answer to this MO question - "Algebraic" topologies like the Zariski topology?

Can one associate a Riemann surface to any arbitrary field extension? The statement there says its true, it is the (equivalence class of) valuations which fix the base field. The result seems too fascinating to be true. Are there any extra hypotheses?

I am aware of the result that the valuations of C(x)/C form the Riemann sphere (counting the exponent of any (x-a) in each rational function gives a correspondence between points on the complex plane (minus infinity) and valuations.)

Any references?

My question is a doubt I had in the last point to the first answer to this MO question - "Algebraic" topologies like the Zariski topology?

Can one associate a Riemann surface to any arbitrary field extension? The statement there says its true, it is the (equivalence class of) valuations which fix the base field. The result seems too fascinating to be true. Are there any extra hypotheses?

I am aware of the result that the valuations of C(x)/C form the Riemann sphere (counting the exponent of any (x-a) in each rational function gives a correspondence between points on the complex plane (minus infinity) and valuations.)

Any references?