Steinitz theorem

Statement

Suppose $k$ is a field and $K$ is a field containing $k$. Then, there exists a subset $T$ of $K$, such that:

• $T$ is an algebraically independent subset of $K$; in other words the field of fractions $k(T)$ embeds inside $K$
• $K$ is algebraic over $k(T)$

Since the extension $k(T)/k$ is purely transcendental and the extension $K/k(T)$ is algebraic, Steinitz theorem can be reformulated as: every field extension can be expressed as an algebraic extension of a purely transcendental extension

Related facts

It is important to note that the order of purely transcendental and algebraic matters: not every field extension can be expressed as a purely transcendental extension of an algebraic extension. In fact, there are examples of extensions where the base field is relatively algebraically closed in the extension, but the extension is not purely transcendental.