Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo.

The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed.

It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry.

The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions.

This introduces a need, in the surface case, to use also valuation theory to describe the phenomena such as blowing up (balloon-style, rather than explosively).

At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman—thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation.