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:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animalsjust as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Devout believers are safeguarded in a high degree against the risk of certain neurotic illnesses; their acceptance of the universal neurosis spares them the task of constructing a personal one.”
—Sigmund Freud (18561939)