Dimension of An Algebraic Variety - Definition By The Transcendence Degree

Definition By The Transcendence Degree

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 coordinate ring of V. Using polynomial equations, it is easy to define sets that have 'mixed dimension': a union of a curve and a plane in space, for example. These fail to be irreducible.