Units and ideal classes
An algebraic integer ε is said to be a unit if 1/ε is an algebraic integer. This is equivalent to the condition Nε = ±1. Indeed the conjugates of an algebraic integer are again algebraic integers, whence, if ε is a unit, then Nε and 1/Nε are rational integers and so ±1. Conversely if Nε = ±1 then 1/ε = ±Nε/ε which is clearly an algebraic integer. The set of all units form a group under multiplication and the set of units in a number field K form a subgroup K. Further, we see that if [α], [β] are principal ideals ...