Present $L=K[t]/(f)$ by sending $t$ to $b$. Applying the <Conormal exact sequence for Kähler differentials> and part ii gives
$$
L\cong(f)/(f^2)\longrightarrow
(\Omega_{K/k}\otimes_KL)\oplus L,dt
\longrightarrow\Omega_{L/k}\longrightarrow0.
$$
Writing $f(t)=\sum_i a_it^i$, the image of $1$, represented by $f$, is
$$
df=\left(\sum_i b^i,da_i, f'(b),dt\right).
$$
This has the required form $( *,f'(b)dt)$.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-126/1/a/solution