Definition:Algebraically Independent

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Let $L / K$ be a field extension.

Let $A \subseteq L$ be a subset of $L$.

Let $\map K {\set {X_\alpha: \alpha \in A} }$ be the field of rational functions in the indeterminates $\family {X_\alpha}_{\alpha \mathop \in A}$.

Then $A$ is algebraically independent over $K$ if and only if there exists a homomorphism:

$\phi: \map K {\set {X_\alpha: \alpha \in A} } \to L$

such that, for all $\alpha \in A$:

$\map \phi {X_\alpha} = \alpha$