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algebraic element
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).
== Examples ==
* The square root of 2 is algebraic over Q, since it is the root of the polynomial ''g''(''x'') = ''x''2 - 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 -π) 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.)
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