In Neukirch "Algebraic number theory" Chapter VII section 7, the concept "ideal numbers" is treated as follows:
, 
.
This is "somewhat forgotten" concept and I have been not familiar with it.
But recently I saw the following description in Köhler "Eta Products and Theta Series Identities" 5.5: 
, 
.
How are these "equivalent"? I ask you to formulate it and prove it. Thank you.
