In

_{''n''} (or $\backslash mu\backslash !\backslash !\backslash mu\_n$). It gives rise to a _{''p''} over a

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the term multiplicative group refers to one of the following concepts:
*the group
A group is a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...

under multiplication of the invertible
In mathematics, the concept of an inverse element generalises the concepts of additive inverse, opposite () and Multiplicative inverse, reciprocal () of numbers.
Given an operation (mathematics), operation denoted here , and an identity element ...

elements of a field, 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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

of ''F'' and the binary operation
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

• is the field multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

,
*the algebraic torus GL(1)..
Examples

*The multiplicative group of integers modulo ''n'' is the group under multiplication of the invertible elements of $\backslash mathbb/n\backslash mathbb$. When ''n'' is not prime, there are elements other than zero that are not invertible. * The multiplicative group ofpositive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...

$\backslash mathbb^+$ is an abelian group
In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...

with 1 its identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...

. The logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

is a group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two ...

of this group to the additive group
An additive group is a group (mathematics), group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian group, abelian, and typically written using the symbol + for its binary operation.
This termin ...

of real numbers, $\backslash mathbb$.
* The multiplicative group of a field $F$ is the set of all nonzero elements: $F^\backslash times\; =\; F\; -\backslash $, under the multiplication operation. If $F$ is finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation) Finite number may refer to:
* A countable number less than infinity, being the cardinality of a finite set – i.e., some natural number
In mathematics, the ...

of order ''q'' (for example ''q'' = ''p'' a prime, and $F\; =\; \backslash mathbb\; F\_p=\backslash mathbb\; Z/p\backslash mathbb\; Z$), then the multiplicative group
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, ring (mathematics), ring, or ...

is cyclic: $F^\backslash times\; \backslash cong\; C\_$.
Group scheme of roots of unity

The group scheme of ''n''-throots of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...

is by definition the kernel of the ''n''-power map on the multiplicative group GL(1), considered as a group scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...

. 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
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field (mathema ...

, with the morphism ''e'' that serves as the identity.
The resulting group scheme is written μreduced scheme
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry.
...

, when we take it over a field ''K'', if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...

the characteristic of ''K'' does not divide ''n''. This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent element
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

s in their structure sheaves); for example μfinite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

with ''p'' elements for any prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...

''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 In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction (field theory), adjunction of ''n''th roots of elements of the base field (mathematics), field. The theory was ...

.
Notes

References

*Michiel Hazewinkel
Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, da ...

, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004.
See also

*Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to ''n'' from the set \ of ''n'' non-negative integers form a group (mathematics), group under multiplication integers modulo n, modulo ''n'', called the multiplicative group of intege ...

*Additive group
An additive group is a group (mathematics), group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian group, abelian, and typically written using the symbol + for its binary operation.
This termin ...

{{DEFAULTSORT:Multiplicative Group
Algebraic structures
Group theory
Field (mathematics)