Idelic modulus ($\Vert x\Vert ={\prod }_v|x_v{|}_v$)

The idelic modulus is the product of normalized local moduli: ordinary real modulus, squared complex modulus and $(N\mathfrak{p}{)}^{-{\mathrm{ord}}_{\mathfrak{p}}(x)}$ at finite places. This continuous homomorphism to ${\mathbb{R}}_{>0}$ has the diagonal multiplicative group in its kernel by the product formula. It differs from the idele norm, which maps between idele groups of field extensions.

The norm-one idele group is the kernel of the idelic modulus. Its quotient by the diagonal $K^{\times }$ is compact for a number field and surjects onto the ideal class group.