Let be a field, and be an algebraically closed extension. An affine algebraic set is the set of the common zeros in of the elements of an ideal in a polynomial ring formula_1 Let formula_2 be the algebra of the polynomial functions over . The dimension of is any of the following integers. It does not change if is enlarged, if is replaced by another algebraically closed extension of and if is replaced by another ideal having the same zeros (that is having the same radical). The dimension is also independent of the choice of coordinates; in other words it does not change if the are replaced by linearly independent linear combinations of them. The dimension of is