Algebraic Independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.
In particular, a one element set {α} is algebraically independent over K if and only if α is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S.
Read more about Algebraic Independence: Example, Algebraic Independence of Known Constants, Lindemann-Weierstrass Theorem, Algebraic Matroids
Famous quotes containing the words algebraic and/or independence:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“I am savage enough to prefer the woods, the wilds, and the independence of Monticello, to all the brilliant pleasures of this gay capital [Paris].”
—Thomas Jefferson (1743–1826)