Galois Extensions of The Rationals
If K is a Galois extension of Q, the theory of Artin L-functions applies to . It has one factor of the Riemann zeta function, which has a pole of residue one, and the quotient is regular at s = 1. This means that the right-hand side of the class number formula can be equated to a left-hand side
- Π L(1,ρ)dim ρ
with ρ running over the classes of irreducible non-trivial complex linear representations of Gal(K/Q) of dimension dim(ρ). That is according to the standard decomposition of the regular representation.
Read more about this topic: Class Number Formula
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