Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 4 / 2 / b

Table of contents
- Solution b

Solution

For each place

v

of

K

and each

w ∣ v

of

L

, the inclusion

K_{v} ↪ L_{w}

i_{L / K} : I_{K} ⟶ I_{L}, (x_{v})_{v} ⟼ (x_{v})_{w ∣ v} .

At all but finitely many finite

v

, the component

x_{v}

is a unit, and its image is a unit at every

w ∣ v

, so the image is an idele. The map is a homomorphism and is injective because every local inclusion is injective.

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