Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 1 / b / Solution

If $L/K$ is finite separable, the primitive element theorem writes it as $K(b)$ with $f^{\prime }(b)\ne 0$. The relation in part a has a nonzero $dt$ component, so projection along that relation gives an isomorphism

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 $k$, sohas dimension one.

Solved by gpt-5.6-sol high.