Affine variety

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, an affine variety or affine algebraic variety is a certain kind of <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> that can be described as a subset of an <a href="/facts/Affine_space/tlvI3oX1">affine space</a>.
More formally, an affine algebraic set is the set of the common <a href="/facts/Zero_of_a_function/mlK3dhoT">zeros</a> over an <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a> k of some family of <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> in the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> 
 
 
 
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 {\displaystyle k[x_{1},\ldots ,x_{n}].}
 
 An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the <a href="/facts/Radical_of_an_ideal/GYm3Pm9C">radical</a> of) the <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> generated by the defining polynomials is <a href="/facts/Prime_ideal/pmrGf6DA">prime</a>. One-dimensional affine varieties are called affine <a href="/facts/Algebraic_curves/LnWsPmDP">algebraic curves</a>, while two-dimensional ones are affine <a href="/facts/Algebraic_surface/2zHU0o9J">algebraic surfaces</a>.
Some texts use the term variety for 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, <a href="/facts/Hilbert%2527s_Nullstellensatz/TlgoRHuM">Hilbert's Nullstellensatz</a>), 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 algebraic 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 rational over k. In the common case where k is the field of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, a k-rational point is called a real point. When the field k is not specified, a rational point is a point that is rational over the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>. For example, <a href="/facts/Fermat%2527s_Last_Theorem/fHRc2t8z">Fermat's Last Theorem</a> asserts that the affine algebraic variety (it is a curve) defined by xn + yn − 1 = 0 has no rational points for any integer n greater than two.

Affine variety open-in-new

Affine variety