Algebraic geometry is the study of solutions of polynomial equations by means of abstract algebra, and in particular ring theory. Algebraic geometry is most easily done over algebraically closed fields, but it can also be done more generally over any field or even over rings.
Affine Algebraic Varieties
One of the first basic objects studied in algebraic geometry is a variety. Let denote affine -space, i.e. a vector space of dimension over an algebraically closed field, such as the field of complex numbers. (We can think of this as -dimensional "complex Euclidean" space.) Let be the polynomial ring in variables, and let be a maximal ideal of . Then is called an affine algebraic variety.
Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties.
(See above remark.)
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