Algebraic geometry is the study of systems of algebraic equations in several variables, and of the structure which one can give to the solutions of such equations. There are four ways in which this study can be carried out: analytic, topological, algebraico-geometric, and arithmetic. It almost goes without saying that these four ways are by no means independent of each other, although each one can be pushed forward by methods appropriate to its point of view.
To use analytic and topological methods, one starts with equations whose coefficients are complex numbers. One may then consider the set of zeros of the equations as a manifold, topological, or analytic, provided one makes suitable assumptions of non-singularity.
The algebraico-geometric methods are applied in dealing with equations having coefficients in an arbitrary field, the solutions of the equations being taken to lie in its algebraic closure, or in a "universal domain." The arguments used are geometric, and are supplemented by as much algebra as the taste of the geometer will allow.