Algebraic extension

In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every element of L is a root of a non-zero polynomial with coefficients in K.^[1][2] A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic.^[3][4]

The algebraic extensions of the field ${\displaystyle \mathbb {Q} }$ of the rational numbers are called algebraic number fields and are the main objects of study of algebraic number theory. Another example of a common algebraic extension is the extension ${\displaystyle \mathbb {C} /\mathbb {R} }$ of the real numbers by the complex numbers.