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

Table of contents
- Solution ii

Solution

Every

k

-derivation

D : K [t] \to M

is uniquely determined by its restriction to

K

and by

D (t)

. Conversely, a

k

-derivation

K \to M

and an arbitrary element

m \in M

extend uniquely by

D (\sum_{i} a_{i} t^{i}) = \sum_{i} D (a_{i}) t^{i} + \sum_{i} i a_{i} t^{i - 1} m .

Representing this natural decomposition of derivations gives

Ω_{K [t] / k} ≅ (Ω_{K / k} \otimes_{K} K [t]) \oplus K [t], d t .

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