Multiplicative group

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Multiplicative_group_of_integers_modulo_n/IACS6cgH">multiplicative group of integers modulo n</a> is the group under multiplication of the invertible elements of 
 
 
 
 
 Z
 
 
 /
 
 n
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /n\mathbb {Z} }
 
. When n is not prime, there are elements other than zero that are not invertible.</li>
<li>The multiplicative group of <a href="/facts/Positive_real_numbers/VFXU56H9">positive real numbers</a> 
 
 
 
 
 
 R
 
 
 +
 
 
 
 
 {\displaystyle \mathbb {R} ^{+}}
 
 is an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> with 1 its <a href="/facts/Identity_element/9nss3xHY">identity element</a>. The <a href="/facts/Logarithm/QeXaV97q">logarithm</a> is a <a href="/facts/Group_isomorphism/LC4N1aDw">group isomorphism</a> of this group to the <a href="/facts/Additive_group/djWVpp9D">additive group</a> of real numbers, 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
.</li>
<li>The multiplicative group of a field 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is the set of all nonzero elements: 
 
 
 
 
 F
 
 ×
 
 
 =
 F
 −
 {
 0
 }
 
 
 {\displaystyle F^{\times }=F-\{0\}}
 
, under the multiplication operation. If 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is <a href="/facts/Finite_field/UkMGJo3m">finite</a> of order q (for example q = p a prime, and 
 
 
 
 F
 =
 
 
 F
 
 
 p
 
 
 =
 
 Z
 
 
 /
 
 p
 
 Z
 
 
 
 {\displaystyle F=\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }
 
), then the <a href="/facts/Finite_field/UkMGJo3m">multiplicative group</a> is cyclic: 
 
 
 
 
 F
 
 ×
 
 
 ≅
 
 C
 
 q
 −
 1
 
 
 
 
 {\displaystyle F^{\times }\cong C_{q-1}}
 
.</li></ul>
<h2 id="group-scheme-of-roots-of-unity">Group scheme of roots of unity</h2>
The group scheme of n-th <a href="/facts/Roots_of_unity/EqH0ho1Y">roots of unity</a> is by definition the kernel of the n-power map on the multiplicative group GL(1), considered as a <a href="/facts/Group_scheme/bJW14FE2">group scheme</a>. 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 <a href="/facts/Fiber_product_of_schemes/gSFuGC7l">fiber product of schemes</a>, with the morphism e that serves as the identity.
The resulting group scheme is written μn (or 
 
 
 
 μ
 
 
 
 μ
 
 n
 
 
 
 
 {\displaystyle \mu \!\!\mu _{n}}
 
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>). It gives rise to a <a href="/facts/Reduced_scheme/eh7NLhmS">reduced scheme</a>, when we take it over a field K, <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the <a href="/facts/Characteristic_(field)/D2VzcQaG">characteristic</a> of K does not divide n. This makes it a source of some key examples of non-reduced schemes (schemes with <a href="/facts/Nilpotent_element/rERm2Pp7">nilpotent elements</a> in their <a href="/facts/Structure_sheaf/JyeaTMtX">structure sheaves</a>); for example μp over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> with p elements for any <a href="/facts/Prime_number/L20FLkgn">prime number</a> 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 <a href="/facts/Duality_theory_of_abelian_varieties/xRSRm5KU">duality theory of abelian varieties</a> in characteristic p (theory of <a href="/facts/Pierre_Cartier_(mathematician)/Mo74aNlW">Pierre Cartier</a>). The Galois cohomology of this group scheme is a way of expressing <a href="/facts/Kummer_theory/DWBtcL4A">Kummer theory</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Multiplicative_group_of_integers_modulo_n/IACS6cgH">Multiplicative group of integers modulo n</a></li>
<li><a href="/facts/Additive_group/djWVpp9D">Additive group</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Michiel_Hazewinkel/0E0sx2np">Michiel Hazewinkel</a>, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4020-2690-0</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">See Hazewinkel et al. (2004), p. 2. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Milne, James S. (1980). Étale cohomology. Princeton University Press. pp. xiii, 66. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Multiplicative group open-in-new

Multiplicative group