Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Algebraic element
Summary
In mathematics, 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 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: it is the root of g(x) = x − π, whose coefficients (1 and −pi) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses C/Q, not C/R.)
The following conditions are equivalent for an element of :
is algebraic over ,
the field extension is algebraic, i.e. every element of is algebraic over (here denotes the smallest subfield of containing and ),
the field extension has finite degree, i.e. the dimension of as a -vector space is finite,
where is the set of all elements of that can be written in the form with a polynomial whose coefficients lie in .
To make this more explicit, consider the polynomial evaluation . This is a homomorphism and its kernel is . If 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. The polynomial 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 . Otherwise, is injective and hence we obtain a field isomorphism , where is the field of fractions of , i.e. the field of rational functions on , by the universal property of the field of fractions.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.