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Concept
Affine variety
Summary
In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring k[X_1, \ldots,X_n]. An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime.
Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).
In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the common zeros are considered (that is, the points of the affine algebric set are in Kn). In this case, the variety is said defined over k, and the points of the variety that belong to kn are said k-rational or r
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