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Unit (ring theory)
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=== Ring of integers of a number field === In the ring {{math|'''Z'''[{{sqrt|3}}]}} obtained by adjoining the [[quadratic integer]] {{math|{{sqrt|3}}}} to {{math|'''Z'''}}, one has {{math|1= (2 + {{sqrt|3}})(2 β {{sqrt|3}}) = 1}}, so {{math|2 + {{sqrt|3}}}} is a unit, and so are its powers, so {{math|'''Z'''[{{sqrt|3}}]}} has infinitely many units. More generally, for the [[ring of integers]] {{mvar|R}} in a [[number field]] {{mvar|F}}, [[Dirichlet's unit theorem]] states that {{math|''R''{{sup|Γ}}}} is isomorphic to the group <math display="block">\mathbf Z^n \times \mu_R</math> where <math>\mu_R</math> is the (finite, cyclic) group of roots of unity in {{mvar|R}} and {{mvar|n}}, the [[rank of a module|rank]] of the unit group, is <math display="block">n = r_1 + r_2 -1, </math> where <math>r_1, r_2</math> are the number of real embeddings and the number of pairs of complex embeddings of {{mvar|F}}, respectively. This recovers the {{math|'''Z'''[{{sqrt|3}}]}} example: The unit group of (the ring of integers of) a [[real quadratic field]] is infinite of rank 1, since <math>r_1=2, r_2=0</math>.
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