Transcendental extension

In mathematics, a transcendental extension ${\displaystyle L/K}$ is a field extension such that there exists an element in the field ${\displaystyle L}$ that is transcendental over the field ${\displaystyle K}$; that is, an element that is not a root of any univariate polynomial with coefficients in ${\displaystyle K}$. In other words, a transcendental extension is a field extension that is not algebraic. For example, ${\displaystyle \mathbb {C} }$ and ${\displaystyle \mathbb {R} }$ are both transcendental extensions of ${\displaystyle \mathbb {Q} .}$

A transcendence basis of a field extension ${\displaystyle L/K}$ (or a transcendence basis of ${\displaystyle L}$ over ${\displaystyle K}$) is a maximal algebraically independent subset of ${\displaystyle L}$ over ${\displaystyle K.}$ Transcendence bases share many properties with bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero.

Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic variety is the transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory in positive characteristic a role that is very similar to the role of algebraic number fields in characteristic zero.