Definition 9.26.1. Let $K/k$ be a field extension.

1. A collection of elements $\{ x_ i\} _{i \in I}$ of $K$ is called algebraically independent over $k$ if the map

$k[X_ i; i\in I] \longrightarrow K$

which maps $X_ i$ to $x_ i$ is injective.

2. The field of fractions of a polynomial ring $k[x_ i; i \in I]$ is denoted $k(x_ i; i\in I)$.

3. A purely transcendental extension of $k$ is any field extension $K/k$ isomorphic to the field of fractions of a polynomial ring over $k$.

4. A transcendence basis of $K/k$ is a collection of elements $\{ x_ i\} _{i \in I}$ which are algebraically independent over $k$ and such that the extension $K/k(x_ i; i\in I)$ is algebraic.

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