Content deleted Content added
→Integers modulo m: Add explanation to make it more accessible. |
→Integers modulo m: Correction; not all cyclic groups. |
||
Line 148:
The notation <math>\mathbb{Z}/m\mathbb{Z}</math> is used because this ring is the [[quotient ring]] of <math>\mathbb{Z}</math> by the [[ideal (ring theory)|ideal]] <math>m\mathbb{Z}</math>, the set formed by all multiples of {{math|''m''}}, i.e., all numbers {{math|''k m''}} with <math>k\in\mathbb{Z}.</math>
Under addition, <math>\mathbb Z/m\Z</math> is a [[cyclic group]]
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.
| |||