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.
Read more about this topic: Dimension Of An Algebraic Variety
Famous quotes containing the words definition and/or degree:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“The degree of tolerance attainable at any moment depends on the strain under which society is maintaining its cohesion.”
—George Bernard Shaw (1856–1950)