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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 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).

1 Examples

• 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 numbers R.

2 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 spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for 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 polynomialThe minimal polynomial of an n by n matrix A over a field F is the monic polynomial p ''x over F of least degree such that p ''A 0. Any other non-zero polynomial f with p ''A 0 is a multiple of p''. The following three statements are equivalent: #λ 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 closedAbstract algebra In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F has a zero in F''. In that case, every such polynomial splits into linear factors. It can be shown that a field. The field of complex numbers is an example.