An algebraic set is the common zero set of a collection of polynomials in an affine
space over a field. For example, in 
the circle is the zero
set of 
,
and the point at 
is the common zero set of
and 
. The algebraic set 
is the set
of solutions to 
.
It decomposes into two irreducible algebraic sets, called algebraic
varieties. In general, an algebraic set has a unique irredundant decomposition,
up to order, as a finite union of irreducible algebraic
varieties. No algebraic variety in this
decomposition is contained in another.
The intersection of two algebraic sets is an algebraic set corresponding to the union of their polynomials.
For example, 
and 
intersect
at 
. In fact, an arbitrary intersection
of algebraic sets is an algebraic set. Every finite union
of algebraic sets is algebraic, but an infinite union need
not be. If 
is the set of solutions to 
and 
is the set of solutions to 
, then 
is the set of solutions to
for all 
and 
. Consequently, algebraic sets are the closed
sets in the Zariski topology.
The set of polynomials vanishing on an algebraic set 
is an ideal in the polynomial
ring. Conversely, any ideal defines an algebraic set
since it is a collection of polynomials. Hilbert's
Nullstellensatz describes the precise relationship between ideals
and algebraic sets over an algebraically closed field.
