Transcendental element

Definition. Let $L/K$ be a field extension and $\alpha\in L$. The element $\alpha$ is transcendental over $K$ if

Equivalently, $\alpha$ is not algebraic over K .

If $t$ is transcendental over $K$, the field $K(t)$ of rational functions behaves like a “field of fractions” of the polynomial ring $K[t]$ (compare fraction fields ).

See also. transcendental extension , finitely generated extension .

Examples.

1. If $t$ is an indeterminate, then $t$ is transcendental over any field $K$, and $K(t)$ is the rational function field.
2. The numbers $\pi$ and $e$ are transcendental over $\mathbb{Q}$, hence also transcendental over any subfield of $\mathbb{C}$ contained in $\overline{\mathbb{Q}}$.
3. In $\mathbb{F}_p(t)$, the element $t$ is transcendental over $\mathbb{F}_p$.