### Algebraic Element

__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 .__*Definition :-*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*.

*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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