Multiplicative group
1110742
204021768
2008-04-07T17:04:14Z
Jakob.scholbach
1935000
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{{Basic notions in group theory}}
In [[mathematics]] and [[group theory]] the term '''multiplicative group''' refers to one of the following concepts, depending on the context
*any group <math> \scriptstyle\mathfrak{{G}} \,\!</math> whose [[binary operation]] is written in '''multiplicative notation''' (instead of being written in [[additive notation]] as usual for [[abelian group]]s),
*the '''underlying group under multiplication''' of the [[invertible]] elements of a [[field (mathematics)|field]], [[ring (mathematics)|ring]], or other structure having multiplication as one of its operations. In the case of a field ''F'', the group is {''F'' - {0}, •}, where 0 refers to the [[zero element]] of the ''F'' and the [[binary operation]] • is the field [[multiplication]],
*the [[algebraic torus]] <math> \scriptstyle\mathbf{GL}_1 </math>.
==Group scheme of roots of unity==
The '''group scheme of <math>n</math>-th [[roots of unity]]''' is by definition the kernel of the <math>n</math>-power map on the multiplicative group <math> \scriptstyle\mathbf{GL}_1 </math>, considered as a [[group scheme]]. That is, for any integer <math>n>1</math> we can consider the morphism on the multiplicative group that takes <math>n</math>-th powers, and take an appropriate [[fiber product]] in the sense of [[scheme theory]] of it, with the morphism <math>e</math> that serves as the identity.
The resulting group scheme is written ''<math>\mu_n</math>''. It gives rise to a [[reduced scheme]], when we take it over a field <math> \scriptstyle\mathbb{K} </math>, [[if and only if]] the [[characteristic (field)|characteristic]] of <math> \scriptstyle\mathbb{K} </math> does not divide <math>n</math>. 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 ''<math>\mu_p</math>'' over a [[finite field]] with <math>p</math> elements for any [[prime number]] <math>p</math>.
This phenomenon is not easily expressed in the classical language of algebraic geometry. It turns out to be of major importance, for example, in expressing the [[duality theory of abelian varieties]] in characteristic <math>p</math> (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]]
[[Category:Abstract algebra]]
[[Category:Group theory]]
[[Category:Field theory]]
[[pl:Grupa multiplikatywna]]