Transcendental extension

A field extension $E/F$ is called a transcendental extension if it is not an algebraic extension . Equivalently, $E/F$ is transcendental iff there exists some $\alpha\in E$ that is a transcendental element over $F$.

Thus “transcendental” is an existence condition: it is enough for $E$ to contain one transcendental element over $F$. (Stronger notions such as “purely transcendental” impose additional structure, but are not part of the basic definition.)

Examples

1. The rational function field $F(t)$ is transcendental over $F$: the element $t$ is transcendental over $F$, so $F(t)/F$ is a simple transcendental extension.
2. $\mathbb{C}/\mathbb{Q}$ is transcendental, since $\mathbb{C}$ contains elements (e.g. $\pi$) that are transcendental over $\mathbb{Q}$.
3. $\mathbb{Q}(t,\sqrt2)/\mathbb{Q}$ is transcendental: it contains $t$, which is transcendental over $\mathbb{Q}$, even though $\sqrt2$ is algebraic over $\mathbb{Q}$.