Saturday, 23 November 2013

Algebraic Element

Definition :-  Let   K   be an extension of a field    F ,     then an element   a   which is element of   K    is said to be Algebraic over   F  if there is a non - zero polynomial  p(x)  is element of  F[x]   for which    p(a) = 0 .

 If L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g(x) with coefficients in K such that g(a)=0. Elements of L which are not algebraic over K are called transcendental over K.
These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, C being the field of complex numbers and Q being the field of rational numbers).

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 Element
Definition :-  Let   K    be an extension over  F   then an element  a   which is an element of    K    is said to be Transcendental over   F    if it is not Algebraic over   F    .

Hence we can say that  A complex number is said to be an Algebraic Number if it is Algebraic over the field of Rational Numbers . In other words it means that it satisfies as Polynomial Equation with rational coefficients not all zero .
                        A complex number which is not algebraic is called transcendental  e.g.    Pi    and   e     are transcendental numbers .

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