Algebraic element

In mathematics, if A is an associative algebra over K, then an element a of A is an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial

[1] Elements of A that are not algebraic over K are transcendental over K. A special case of an associative algebra over

is an extension field

These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, with C being the field of complex numbers and Q being the field of rational numbers).

The following conditions are equivalent for an element

of an extension field

: To make this more explicit, consider the polynomial evaluation

This is a homomorphism and its kernel is

is algebraic, this ideal contains non-zero polynomials, but as

is a euclidean domain, it contains a unique polynomial

with minimal degree and leading coefficient

, which then also generates the ideal and must be irreducible.

is called the minimal polynomial of

and it encodes many important properties of

Hence the ring isomorphism

obtained by the homomorphism theorem is an isomorphism of fields, where we can then observe that

is injective and hence we obtain a field isomorphism

is the field of fractions of

, i.e. the field of rational functions on

, by the universal property of the field of fractions.

We can conclude that in any case, we find an isomorphism

Investigating this construction yields the desired results.

This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over

also yields a finite extension, and therefore these elements are algebraic as well.

is a field that sits in between

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.

is algebraically closed, then the field of algebraic elements of

is algebraically closed, which can again be directly shown using the characterisation of simple algebraic extensions above.

An example for this is the field of algebraic numbers.