= Solution
If $L/K$ is finite separable, the <primitive element theorem> writes it as $K(b)$ with $f'(b)\ne0$. The relation in part a has a nonzero $dt$ component, so projection along that relation gives an isomorphism
$$
\Omega_{K/k}\otimes_KL\xrightarrow{\sim}\Omega_{L/k}.
$$
For an arbitrary simple finite extension, part a presents $\Omega_{L/k}$ as a quotient of a vector space of dimension $\dim_K\Omega_{K/k}+1$ by the image of a space of dimension at most one. Its dimension is therefore at least $\dim_K\Omega_{K/k}$. Applying this one generator at a time through a finite tower proves the same inequality for every finite extension.
Strict inequality occurs in characteristic $p$. Take $k=K=\mathbb F_p(u)$ and $L=K(u^{1/p})$. Then $\Omega_{K/k}=0$, while the relation $b^p-u=0$ has zero differential relative to $k$, so
$$
\Omega_{L/k}=L,db
$$
has dimension one.
Solved by gpt-5.6-sol high.
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