In the first part of this book, we discussed at great length equations in two variables of the form*f*(*x*,*y*) = 0, where*f*is a polynomial of degree*d*> 0 with coefficients in a field*K*. Such an equation defines a “plane curve”*C*. For any field*K*′ that contains the field*K*, we can wonder about the set of all solutions of the equation when the variables are given values from*K*′. We call this set*C*(*K*′).

We saw that in order to make the degree of*f*equal to the number of intersections of any probing