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