Multiplicative Group

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## Examples

## Group scheme of roots of unity

## Notes

## References

## See also

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Multiplicative Group

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

- the
**group under multiplication**of the invertible elements of a field,^{[1]}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 , where 0 refers to the zero element of*F*and the binary operation o is the field multiplication, - the algebraic torus GL(1).
^{[clarification needed]}.

- The multiplicative group of integers modulo
*n*is the group under multiplication of the invertible elements of . When*n*is not prime, there are elements other than zero that are not invertible. - The multiplicative group of positive real numbers 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, .
- The multiplicative group of a field is the set of all nonzero elements: , under the multiplication operation. If is finite of order
*q*(for example*q*=*p*a prime, and ), then the multiplicative group is cyclic: .

The **group scheme of n-th roots of unity** is by definition the kernel of the

The resulting group scheme is written ?_{n} (or ^{[2]}). It gives rise to a reduced scheme, when we take it over a field *K*, if and only if the characteristic of *K* does not divide *n*. This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent elements in their structure sheaves); for example ?_{p} 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). The Galois cohomology of this group scheme is a way of expressing Kummer theory.

- Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikha?lovna Gubareni, Vladimir V. Kirichenko.
*Algebras, rings and modules*. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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