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. That means that any two rational functions F and G on it satisfy some polynomial relation P(F,G) = 0. That implies that F and G are constrained to take related values (up to some finite freedom of choice): they cannot be truly independent.
### Formal definition For an algebraic variety *V* over a field *K*, the **dimension** of *V* is the transcendence degree over *K* of the function field *K(V)* of all rational functions on *V*, with values in *K*. For the function field even to be defined, *V* here must be an irreducible algebraic set; in which case the function field (for an affine variety) is just the field of fractions of the co-ordinate ring of *V*. It is easy to define by polynomials sets that have 'mixed dimension': a union of a curve and a plane in space, for example. These fail to be irreducible. |