Editing Multiplicative group

{{Short description|Mathematical structure with multiplication as its operation}}
{{Group theory sidebar |Basics}}

In [[mathematics]] and [[group theory]], the term '''multiplicative group''' refers to one of the following concepts:

*the '''[[group (mathematics)|group]] under multiplication''' of the [[invertible]] elements of a [[field (mathematics)|field]],<ref>See Hazewinkel et al. (2004), p. 2.</ref> [[ring (mathematics)|ring]], or other structure for which one of its operations is referred to as multiplication. In the case of a field ''F'', the group is {{nowrap|(''F'' ∖ {0}, •)}}, where 0 refers to the [[zero element]] of ''F'' and the [[binary operation]] • is the field [[multiplication]],
*the [[algebraic torus]] GL(1).{{clarify|reason=this is not defined in this article nor in the linked article|date=March 2015}}

== Examples ==
*The [[multiplicative group of integers modulo n|multiplicative group of integers modulo ''n'']] is the group under multiplication of the invertible elements of <math>\mathbb{Z}/n\mathbb{Z}</math>.  When ''n'' is not prime, there are elements other than zero that are not invertible.
* The multiplicative group of [[positive real numbers]] <math>\mathbb{R}^+</math> is an [[abelian group]] with 1 its [[identity element]]. The [[logarithm]] is a [[group isomorphism]] of this group to the [[additive group]] of real numbers, <math>\mathbb{R}</math>.
* The multiplicative group of a field <math>F</math> is the set of all nonzero elements: <math>F^\times = F -\{0\}</math>, under the multiplication operation. If <math>F</math> is [[finite field|finite]] of order ''q'' (for example ''q'' = ''p'' a prime, and <math>F = \mathbb F_p=\mathbb Z/p\mathbb Z</math>), then the [[finite field#Multiplicative_structure|multiplicative group]] is cyclic: <math>F^\times \cong C_{q-1}</math>.

==Group scheme of roots of unity==
The '''group scheme of ''n''-th [[roots of unity]]''' is by definition the kernel of the ''n''-power map on the multiplicative group GL(1), considered as a [[group scheme]]. That is, for any integer ''n'' > 1 we can consider the morphism on the multiplicative group that takes ''n''-th powers, and take an appropriate [[fiber product of schemes]], with the morphism ''e'' that serves as the identity.

The resulting group scheme is written μ<sub>''n''</sub> (or <math>\mu\!\!\mu_n</math><ref>{{cite book | last=Milne | first=James S. | title=Étale cohomology | publisher=Princeton University Press | year=1980 | pages=xiii, 66 }}</ref>). It gives rise to a [[reduced scheme]], when we take it over a field ''K'', [[if and only if]] the [[characteristic (field)|characteristic]] of ''K'' does not divide ''n''. This makes it a source of some key examples of non-reduced schemes (schemes with [[nilpotent element]]s in their [[structure sheaf|structure sheaves]]); for example μ<sub>''p''</sub> over a [[finite field]] with ''p'' elements for any [[prime number]] ''p''. 

This phenomenon is not easily expressed in the classical language of algebraic geometry. For example, it turns out to be of major importance in expressing the [[duality theory of abelian varieties]] in characteristic ''p'' (theory of [[Pierre Cartier (mathematician)|Pierre Cartier]]). The [[Galois cohomology]] of this group scheme is a way of expressing [[Kummer theory]].

== See also ==
*[[Multiplicative group of integers modulo n]]
*[[Additive group]]

== Notes ==
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== References ==
* [[Michiel Hazewinkel]], Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004. {{isbn|1-4020-2690-0}}

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[[Category:Algebraic structures]]
[[Category:Group theory]]
[[Category:Field (mathematics)]]