Axiomatic WikiWWikiDDefinitionsThe units modulo n
Definition·D117
The residue classes modulo n that have a multiplicative inverse.
For
, the units modulo
are the residue classes with a multiplicative inverse:
with
from multiplication of classes and
the class of
(D114).
In words
The units modulo n are those residue classes u having a class v whose product with u is the class of 1: the classes with a multiplicative inverse.
Never needed: F03 · F04 · F05 · F06 · F08 · F10 · F11 · F12 · F13 · A03 · A04 · A05 · A07 · A08 · A09 (computed from the citation graph, not asserted).
Remarks
By T59,
is a commutative ring with unity
, so a unit is an invertible element in the usual ring sense. Because multiplication is commutative, the one-sided condition
already gives a two-sided inverse (
too), and that inverse
is unique. The units form a group under multiplication (T61), and L78 identifies them exactly:
is a unit precisely when
and
are coprime. Their number is Euler's totient
. Examples: modulo
the units are
; modulo
they are
. When
is prime every nonzero class is a unit.