Editing Adele ring (section)

===Norm on the idele group===

The trace and the norm should be transfer from the adele ring to the idele group. It turns out the trace can't be transferred so easily. However, it is possible to transfer the norm from the adele ring to the idele group. Let <math>\alpha \in I_K.</math> Then <math>\operatorname{con}_{L/K}(\alpha) \in I_L</math> and therefore, it can be said that in injective group homomorphism

:<math>\operatorname{con}_{L/K}: I_K \hookrightarrow I_L.</math>

Since <math>\alpha \in I_L,</math> it is invertible, <math>N_{L/K}(\alpha)</math> is invertible too, because <math>(N_{L/K}(\alpha))^{-1}= N_{L/K}(\alpha^{-1}).</math> Therefore <math>N_{L/K}(\alpha) \in I_K.</math> As a consequence, the restriction of the norm-function introduces a continuous function:

:<math>N_{L/K}: I_L \to I_K.</math>