Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 3 / c

Table of contents
- Solution c

Solution

By the Kronecker–Weber theorem, choose

m

with

K \subseteq Q (ζ_{m}) .

Restriction gives a quotient

(Z / m Z)^{\times} ↠ G = Gal (K / Q) .

Every character

ψ \in G

inflates along this quotient and has an associated primitive Dirichlet character

χ_{ψ}

, whose conductor may divide

m

. Comparing Euler factors, or applying the factorization of the Artin

L

-function of the regular representation, gives

ζ_{K} (s) = ψ \in G \prod L (s, χ_{ψ}) .

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