# Steinitz theorem

From Commalg

## Statement

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

- is an algebraically independent subset of ; in other words the field of fractions embeds inside
- is algebraic over

Since the extension is purely transcendental and the extension 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.