The operation just described can be proved to satisfy the group properties. It can also be written out using rational functions in the coefficients and coordinates. Note that when the field is not R we may no longer be able to draw nice graphs but the rational functions defining the group operation still make sense and can be used to compute in the group. We will be concerned with the case when the field is a finite field. Note that the group of points on the curve will necessarily be finite too.
Typical examples are rational numbers, real numbers or complex numbers.
An example of a finite field is the set of integers modulo a prime integer. The simplest possible one is the integers modulo 2 i.e., the booleans. Addition and multiplication are XOR and AND respectively. Another way is to take a small prime, such as 2, and do a field extension. For an example of a field extension, start with the boolean field just described.