Purely transcendental extension

This article defines a field extension property: a property that can be evaluated to true/false for any field extension.
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Definition

Suppose $L$ is a field extension of a field $K$. We say that $L$ is purely transcendental if there exists a subset $S$ of $L$ such that $S$ generates $L$ over $K$ and $S$ is algebraically independent over $K$: in other words, for any polynomial $p\in K[x_{1},x_{2},\dots ,x_{n}]$, and any distinct $s_{1},s_{2},\dots ,s_{n}\in S$, $p(s_{1},s_{2},\dots ,s_{n})\neq 0$.

Such a subset $S$ is termed a transcendence base.

Relation with other properties

Opposite properties

• Algebraic extension

Facts

• Steinitz theorem states that every field extension is an algebraic extension of a purely transcendental extension.
• Luroth's theorem states that any sub-extension of a purely transcendental extension with transendence base of size one is also purely transcendental.