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:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“When I tried to talk to my father about the kind of work I might do after college, he said, “You know, Charlotte, I’ve been giving a lot of thought to that, and it seems to me that the world really needs good, competent secretaries. Your English degree will help you.” He said this with perfect seriousness. I was an A student at Bryn Mawr ...”
—Charlotte Palmer (b. c. 1925)