Field Theory [Lecture notes]

Description

Field TheoryThere is, I believe, a finite amount of "general field theory". Most graduate level algebra courses concentrate on the structure theory of algebraic field extensions, which is of course both beautiful and useful. However, many algebraists need to know more than this. Speaking as an arithmetic algebraic geometer, the structure theory of transcendental extensions -- and especially, the notion of a separable transcendental extension -- inevitably comes up, as do certain other constructions which don't seem to make it into the standard course: e.g. linear disjointness, composita, tensor products of fields, a systematic treatment of norms and traces. The notes which follow aim to be a "serious" account of all aspects of general field theory. At the moment they cover about half of the material that they should, unfortunately for the most part the better known half. (Update: now about 62%!) Moreover what is present is quite rough: please consider it only a first draft. Early Draft (pdf) (110 pages)