Abstract Algebraic Variety - Examples - Affine Algebraic Variety - Example 2

Example 2

Let again k be the field of complex numbers C. Let A2 be a two dimensional affine space over C. The polynomials g in the ring C can be viewed as complex valued functions on A2 by evaluating g at the points in A2. Let subset S of C contain a single element g(x, y):

The zero-locus of g(x, y) is the set of points in A2 on which this function vanishes, that is the set of points (x,y) such that x2 + y2 = 1. As g(x, y) is an absolutely irreducible polynomial, this is an algebraic variety. The set of its real points (that is the points for which x and y are real numbers), is known as the unit circle; this name is also often given to the whole variety.