Analogy With Vector Space Dimensions
There is an analogy with the theory of vector space dimensions. The dictionary matches algebraically independent sets with linearly independent sets; sets S such that L is algebraic over K(S) with spanning sets; transcendence bases with bases; and transcendence degree with dimension. The fact that transcendence bases always exist (like the fact that bases always exist in linear algebra) requires the axiom of choice. The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma.
This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of matroids, called linear matroids and algebraic matroids respectively. Thus, the transcendence degree is the rank function of an algebraic matroid. Every linear matroid is isomorphic to an algebraic matroid, but not vice versa.
Read more about this topic: Transcendence Degree
Famous quotes containing the words analogy, space and/or dimensions:
“The whole of natural theology ... resolves itself into one simple, though somewhat ambiguous proposition, That the cause or causes of order in the universe probably bear some remote analogy to human intelligence.”
—David Hume (1711–1776)
“I would have broke mine eye-strings, cracked them, but
To look upon him, till the diminution
Of space had pointed him sharp as my needle;
Nay, followed him till he had melted from
The smallness of a gnat to air, and then
Have turned mine eye and wept.”
—William Shakespeare (1564–1616)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)