Algebraic Element

Definition

Given a field extension K/F, an element αK is called algebraic over F if it is the root of some polynomial in F[X]{0}.

Intuitively, this is saying that there is a non-zero polynomial which pulls α into F.

If an element is not algebraic it is called transcendental.

Trivially, any element is algebraic over field containing it.

For non-trivial examples, consider that iC is algebraic over R due to the polynomial X2+1, and 2R is algebraic over Q due to the polynomial X22.