Wikipedia / Algebraic element
Wikipedia / Algebraic element
In mathematics, if is an associative algebra over , then an element of is an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of that are not algebraic over are transcendental over is an extension field is called an algebraic extension.
These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is , with being the field of complex numbers and being the field of rational numbers).
Examples
The square root of 2 is algebraic over , since it is the root of the polynomial whose coefficients are rational.
Pi is transcendental over but algebraic over the field of real numbers : it is the root of , whose coefficients (1 and −) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses , not .)
An indeterminate x in a field K(x) of rational functions or K((x)) of formal Laurent series is defined to be transcendental over K.
Algebraic functions are the algebraic elements over a field of rational functions K(x1, ..., xm).
Properties
:
,
is algebraic, i.e. every),
has finite degree, i.e. the dimension-vector space is finite,
.
To make this more explicit, consider the polynomial evaluation. This is a homomorphism and its kernel is algebraic, this ideal is a euclidean domain, which then also generates the ideal and must be irreducible is called the minimal polynomial obtained by the homomorphism theorem is the field of fractions, i.e. the field of rational functions. Investigating this construction yields the desired results.
.
Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed 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.
See also
References
Further reading