Transcendental field extension

Definition. A field extension $L/K$ is transcendental if there exists $\alpha\in L$ that is transcendental over K . Equivalently, $L/K$ is transcendental if it is not an algebraic extension .

A basic class of examples is the rational function field $K(t)$, obtained by adjoining a transcendental element $t$.

See also. finitely generated extensions , simple extension .

Examples.

1. $K(t)/K$ is transcendental, with $t$ transcendental over $K$.
2. $\mathbb{R}/\mathbb{Q}$ is transcendental, since $\mathbb{R}$ contains transcendental numbers (e.g. $\pi$).
3. $\mathbb{F}_p(t)/\mathbb{F}_p$ is transcendental.