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A Comprehensive Course in Number Theory by Alan Baker

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

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