Editing Algebraic number theory (section)

===Dirichlet's unit theorem===
{{Main|Dirichlet's unit theorem}}
Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units ''O''<sup>×</sup> of the ring of integers ''O''. Specifically, it states that ''O''<sup>×</sup> is isomorphic to ''G'' × '''Z'''<sup>''r''</sup>, where ''G'' is the finite [[cyclic group]] consisting of all the roots of unity in ''O'', and ''r'' = ''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>&nbsp;−&nbsp;1 (where ''r''<sub>1</sub> (respectively, ''r''<sub>2</sub>) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of ''K''). In other words, ''O''<sup>×</sup> is a [[finitely generated abelian group]] of [[Rank of an abelian group|rank]] ''r''<sub>1</sub>&nbsp;+&nbsp;''r''<sub>2</sub>&nbsp;−&nbsp;1 whose torsion consists of the roots of unity in ''O''.