In mathematics, the roots of polynomials are in abstract algebra called algebraic elements. They can be created in a larger structure ('adjoined'), not simply found to exist in a given one.
More precisely, if L is a field extension of K then an element a of L is called an algebraic element over K, or just algebraic overK, if there exists some non-zero polynomialg(x) with coefficients in K such that g(a)=0. Elements of L which are not algebraic over K are called transcendental over K.
The square root of 2 is algebraic over Q, since it is the root of the polynomial g(x) = x2 - 2 whose coefficients are rational.
Pi is transcendental over Q but algebraic over the field of real numbersR.
Properties
The following conditions are equivalent for an element a of L:
a is algebraic over K
the field extension K(a)/K has finite degree, i.e. the dimension of K(a) as a K-vector space is finite. (Here K(a) denotes the smallest subfield of L containing K and a)
K[a] = K(a), where K[a] is the set of all elements of L that can be written in the form g(a) with a polynomial g whose coefficients lie in K.
This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K. The set of all elements of L which are algebraic over K is a field that sits in between L and K.
If a is algebraic over K, then there are many non-zero polynomials g(x) with coefficients in K such that g(a) = 0. However there is a single one with smallest degree and with leading coefficient 1. This is the minimal polynomial of a and it encodes many important properties of a.
Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed. The field of complex numbers is an example.
Transcendental Meditation (TM) is practised for fifteen to twenty minutes in the morning and evening, while sitting comfortably with the eyes closed.
The Transcendental Meditation Programme is the most direct way for the fulfilment of the supreme philosophy of life, where the individual lives life in full enlightenment, and where the individual breathes life in Cosmic Reality.
Transcendental Meditation helps the awareness to transcend boundaries and go beyond the field of limitations; this is how Transcendental Meditation, providing the opportunity to create unboundedness, infinity, eternity, satisfies the inner creativity and inner genius of life by providing an opportunity for the full expression of Creative Intelligence.
Transcendentalism was not only a literary, philosophical, and religious movement; it was also, inescapably, a social and political movement as well.
In religion transcendentalism teaches that the religious spirit is a necessary aspect of human nature--or of the human condition--and that the religious spirit does not reside in external forms, words, ceremonies, or institutions.
Transcendentalism's commitment ot the individual and to the principle of individuation is a commitment to the soul or spirit that each person possesses in common with all other human beings.
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