= Differentials detect algebraic independence in characteristic zero
For a finitely generated <field extension> $L/k$ in characteristic zero, a tuple $t_i$ is algebraically independent exactly when its differentials $dt_i$ are linearly independent in the <Kähler differentials>. A relation of minimum total degree gives a nontrivial differential relation; an independent tuple extends to a <transcendence basis>. In positive characteristic, derivatives of nonconstant relations can vanish identically.
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