Algebraic set

In mathematics, an algebraic set over a field "K" is the set of solutions in "K"^"n" ("n"-tuples of elements of "K") of a set of simultaneous equations

:"P"₁("X"₁, ...,"X"_"n") = 0:"P"₂("X"₁, ...,"X"_"n") = 0

and so on up to

:"P"_"m"("X"₁, ...,"X"_"n") = 0

for some integer "m", where the "P"_i are polynomials over "K". That is, we consider the simultaneous solution set of these equations applied to vectors

:("x"₁, ...,"x"_"n")

with the "x"_i taken from "K".

Algebraic sets are the primitive objects of algebraic geometry. To get the standard concept of algebraic variety, however, two extra aspects need to be introduced:

*"K" should be an algebraically closed field, for example the complex numbers.
*The irreducible sets are the fundamental objects.

Under these two conditions there is a satisfactory definition of dimension. Also, if "K" is the real number field, an algebraic set can easily be the empty set in cases where the complex number solutions are numerous.

References

*cite book
author = Robin Hartshorne
year = 1997
title = Algebraic Geometry
publisher = Springer-Verlag
id = ISBN 0-387-90244-9
* Citation
last=Milne
first=James S.
author-link=James S. Milne
title=Algebraic Geometry
year=2008
url=http://www.jmilne.org/math/CourseNotes/math631.html
accessdate=2008-07-16