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Inmathematics andgroup theory, the termmultiplicative group refers to one of the following concepts:

• thegroup under multiplication of theinvertible elements of afield,^[1]ring, or other structure for which one of its operations is referred to as multiplication. In the case of a fieldF, the group is(F ∖ {0}, •), where 0 refers to thezero element ofF and thebinary operation • is the fieldmultiplication,
• thealgebraic torus GL(1).^{[clarification needed]}

• Themultiplicative group of integers modulon is the group under multiplication of the invertible elements of${\displaystyle \mathbb{Z}/n\mathbb{Z}}$. Whenn is not prime, there are elements other than zero that are not invertible.
• The multiplicative group ofpositive real numbers${\displaystyle {\mathbb{R}}^{+}}$ is anabelian group with 1 itsidentity element. Thelogarithm is agroup isomorphism of this group to theadditive group of real numbers,${\displaystyle \mathbb{R}}$.
• The multiplicative group of a field${\displaystyle F}$ is the set of all nonzero elements:${\displaystyle F^{\times }=F-\{0\}}$, under the multiplication operation. If${\displaystyle F}$ isfinite of orderq (for exampleq =p a prime, and${\displaystyle F={\mathbb{F}}_p=\mathbb{Z}/p\mathbb{Z}}$), then themultiplicative group is cyclic:${\displaystyle F^{\times }\cong C_{q-1}}$.

Thegroup scheme ofn-throots of unity is by definition the kernel of then-power map on the multiplicative group GL(1), considered as agroup scheme. That is, for any integern > 1 we can consider the morphism on the multiplicative group that takesn-th powers, and take an appropriatefiber product of schemes, with the morphisme that serves as the identity.

The resulting group scheme is written μ_n (or${\displaystyle \mu {\mu }_n}$^[2]). It gives rise to areduced scheme, when we take it over a fieldK,if and only if thecharacteristic ofK does not dividen. This makes it a source of some key examples of non-reduced schemes (schemes withnilpotent elements in theirstructure sheaves); for example μ_p over afinite field withp elements for anyprime numberp.

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 theduality theory of abelian varieties in characteristicp (theory ofPierre Cartier). The Galois cohomology of this group scheme is a way of expressingKummer theory.

• Multiplicative group of integers modulo n
• Additive group

• Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko.Algebras, rings and modules. Volume 1. 2004. Springer, 2004.ISBN 1-4020-2690-0

• Algebraic structures
• Group theory
• Field (mathematics)

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• Wikipedia articles needing clarification from March 2015