 Kähler differentials under a separable field extension (source code)

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= Kähler differentials under a separable field extension
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{title2=$\Omega_{L/k}=L\otimes_F\Omega_{F/k}$}

For finite separable $L/F$, every $k$-<derivation> on $F$ into an $L$-module extends uniquely to $L$. Differentiate the minimal polynomial of each algebraic generator and divide by its nonzero derivative. Consequently the <Transitivity exact sequence for Kähler differentials> identifies $\Omega_{L/k}$ with $L\otimes_F\Omega_{F/k}$, and $\Omega_{L/F}=0$.

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