# Definition:Algebraically Independent

## Definition

Let $L / K$ be a field extension.

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

Let $\map K {\set {X_\alpha}_{\alpha \mathop \in A} }$ be the Field of Rational Functions in the indeterminates $\set {X_\alpha: \alpha \mathop \in A}$.

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

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

such that:

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