Transcendental Element

Definition

Given a field extension \(\mathbb{K}/\mathbb{F}\), an element \(\alpha \in \mathbb{K}\) is called transcendental if it is not the root of any polynomial in \(\mathbb{F}[X] - \{0\}\).

If an element is not transcendental it is called algebraic.

Proving elements are algebraic is a matter of finding the corresponding polynomial (non-trivial, but certainly doable). Proving elements are transcendental often requires much more exotic and varying methods.