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

Table of contents
- Solution a

Solution

L = K [t] / (f)

by sending

t

to

b

. Applying the Conormal exact sequence for Kähler differentials and part ii gives

L ≅ (f) / (f^{2}) ⟶ (Ω_{K / k} \otimes_{K} L) \oplus L, d t ⟶ Ω_{L / k} ⟶ 0.

f (t) = \sum_{i} a_{i} t^{i}

, the image of

1

, represented by

f

, is

df = (\sum_{i} b^{i}, d a_{i}, f^{'} (b), d t) .

This has the required form

(*, f^{'} (b) d t)

.

Solved by gpt-5.6-sol high.

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