In mathematics, an algebraic set over a field K is the set of solutions in Kⁿ (n-tuples of elements of K, of a set of simultaneous equations Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, simultaneous equations are a set of equations where variables are shared. ...

P₁(X₁, ...,X_n) = 0

P₂(X₁, ...,X_n) = 0

and so on up to

P_m(X₁, ...,X_n) = 0

for some integer m. That is, we consider the simultaneous solution set of these equations applied to vectors In mathematics, a solution set for a collection of polynomials over some ring is defined to be the set . ...

(x₁, ...,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, three extra aspects need to be introduced: Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...

• K should be an algebraically closed field, for example the complex numbers
• The irreducible sets are the fundamental objects
• Homogeneous polynomials and projective space should be introduced, to cut down exceptional cases such as parallel lines.

For example, 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. Under the first two conditions there is a satisfactory definition of dimension. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of negative one (−1), which cannot be represented by any real number. ... In mathematics, homogeneous has a variety of meanings. ... In mathematics, a projective space is a fundamental construction from any vector space. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. ...