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2.2. VARIETIES 13
Theorem 2.4. (Hilbert Basis Theorem). Every ideal in K[x 1 ,...,x n ] is finitely generated.
Note that a given ideal may have many different bases. A particular kind of basis
is a Gröbner bases, which has a number of useful additional properties as is shown in
Section 2.4.
Definition 2.5. (Minimal basis). A basis F of an ideal I is said to be minimal if
there exists no proper subset of F which also is a basis of I.
Example 2.2. {x 2 ,x 4 } is not a minimal basis, since x 2 generates x 4 .
A property of an ideal is that it can be added to another ideal. This results, again,
in an ideal: the sum of ideal I and J is the set that contains all the sums of elements
of the ideals I and J. It is denoted by I + J and is given by:
I + J = {i + j :(i, j) ∈ I × J}, (2.1)
where × is the Cartesian product.
2.2 Varieties
An affine algebraic variety is a set which has the property that it is the smallest set
containing all the common zeros of a subset of polynomials from a polynomial ring.
Let X = {x 1 ,...,x n } be a non-empty set of variables, and let S ⊆ K[X]. The variety
of S is the set where each of the polynomials in S becomes zero.
Definition 2.6. (Affine algebraic variety). Let S = {f 1 ,...f m } be polynomials in
K[X]. The affine variety of S in K n is denoted by V (S) and is defined as:
V (S) ={(a 1 ,...,a n ) ∈ K n : f i (a 1 ,...,a n )=0 for all 1 ≤ i ≤ m}. (2.2)
Example 2.3. Let K = R and X = {x 1 ,x 2 }, let S = {f 1 ,f 2 } with f 1 =3x 1 + x 2 +5,
and f 2 = x 1 + x 2 − 1. The intersection of the lines f 1 = 0 and f 2 = 0 in the
two-dimensional real space is in one-to-one correspondence with the set of common
zeros: the variety in R 2 of the ideal I of R[x 1 ,x 2 ] which is generated by 〈f 1 ,f 2 〉.
The following demonstrates how to find this intersection by manipulating with the