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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, if every element of L is a root of a non-zero polynomial with coefficients in K. A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic.
The algebraic extensions of the field Q {\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 C / R {\displaystyle \mathbb {C} /\mathbb {R} } of the real numbers by the complex numbers.
Extension of a mathematical field with polynomial roots