>>11235566I know what you are saying, but the set of real solutions to the equation is just not how you should think about a variety, especially over R. More crucially, that is not how you define a variety over R to get sensible results. And that is not what authors mean by variety over a non-algebraically closed field.
You cannot think of a variety over R as just a subset of R^n or P^n(R), otherwise you will lose a lot of information.
For example, you cannot make the difference between the varieties {x^2+y^2 = 0} and {x= 0, y=0} in the real affine space, whereas they have very different geometric properties. One is a point and the other is a curve, but they both have the same set of real solutions.
You should think about the circle at least as the set of its complex solutions equipped with complex conjugation, or better, as the real algebra R[x,y]/(x^2+y^2-1) or as the equation "x^2+y^2 = 1", but definitely not as just the subset of R^2 {x^2+y^2 = 1}.
