As noted in the question "[Links between Riemann surfaces and algebraic geometry](https://mathoverflow.net/questions/2704/links-between-riemann-surfaces-and-algebraic-geometry)", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces are equivalent to smooth complex algebraic curves.


I wondered if anyone could give insights into the history of this relationship - for example, which was studied first, who first realised the connection, did the theories develop concurrently for any period of time, etc.


Any individual insights, or links or book recommendations to find out more, would be greatly appreciated.



History of the connection between Riemann surfaces and complex algebraic curves
The connection came from the paper by Dedekind and Weber "*Theorie der algebraischen Functionen einer Veranderlichen*", Crelle's Journal, 1882. In this paper the authors recover the theory by Riemann (including the famous Riemann-Roch theorem) by the abstract procedure of assigning a "curve" to an degree one transcendental field extension $\Omega/\mathbb{C}$ where "points" correspond to "places", i.e. discrete rank one valuations of the field $\Omega$. This theory inspired later further developments by Weil and Zariski and it may be considered the prehistory of the modern approach of abstract algebraic geometry.


This is also one of the first instances of modern structural mathematics based on (infinite) sets, much on the style later embraced by Bourbaki. The emphasis is on fields, valuations and ideals, the geometric counterpart of Dedekind's work in algebraic number theory. This point of view was not very well appreciated by some outstanding figures like Kronecker. This may be considered as a prelude of the controversy between formalists and intuitionists (Hilbert vs. Brouwer) that happened a few decades later.


There is an nice translation of Dedekind and Weber's paper by Stillwell published by the AMS, see [AMS bookstore](http://www.ams.org/bookstore-getitem/item=hmath-39).


