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The ring of integers modulo {{math|''m''}} is a [[field (mathematics)|field]], i.e., every nonzero element has a [[Modular multiplicative inverse|multiplicative inverse]], if and only if {{math|''m''}} is [[Prime number|prime]]. If {{math|1=''m'' = ''p''{{i sup|''k''}}}} is a [[prime power]] with {{math|''k'' > 1}}, there exists a unique (up to isomorphism) finite field <math>\mathrm{GF}(m) =\mathbb F_m</math> with {{math|''m''}} elements, which is ''not'' isomorphic to <math>\mathbb Z/m\mathbb Z</math>, which fails to be a field because it has [[zero-divisor]]s.
If {{math|''m'' > 1}}, <math>(\mathbb Z/m\mathbb Z)^\times</math> denotes the [[multiplicative group of integers modulo n|multiplicative group of the integers modulo {{math|''m''}}]] that are invertible. It consists of the congruence classes {{math|{{overline|''a''}}{{sub|''m''}}}}, where {{math|''a''}} [[coprime integers|is coprime]] to {{math|''m''}}; these are precisely the classes possessing a multiplicative inverse. They form an [[abelian group]] under multiplication; its order is {{math|''φ''(''m'')}}, where {{mvar|φ}} is [[Euler's totient function]]
== Applications ==
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