12

Units and ideal classes

**12.1 Units**

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 ...

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