Multiplicative group

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Multiplicative group

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
the term multiplicative group refers to one of the following concepts, depending on the context

Group scheme of roots of unity
The group scheme of -th roots of unity is by definition the kernel of the -power map on the multiplicative group , considered as a group scheme

Group scheme

In mathematics, a group scheme is a group object in the category of schemes. That is, it is a scheme G with the equivalent properties* there is a group law expressible as a multiplication ? and inversion map ? on G; or...
. That is, for any integer we can consider the morphism on the multiplicative group that takes -th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism that serves as the identity.

The resulting group scheme is written '.

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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
the term multiplicative group refers to one of the following concepts, depending on the context

any group whose binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
is written in multiplicative notation (instead of being written in additive notation as usual for abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
s),
the underlying group under multiplication of the invertible elements of a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
, ring
Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
, or other structure having multiplication as one of its operations. In the case of a field F, the group is , where 0 refers to the zero element
Zero element

In mathematics, a zero element is one of several generalizations of 0 to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context:...
of the F and the binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
• is the field multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
,
the algebraic torus
Algebraic torus

In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory ....
.

Group scheme of roots of unity

The group scheme of -th roots of unity is by definition the kernel of the -power map on the multiplicative group , considered as a group scheme
Group scheme

In mathematics, a group scheme is a group object in the category of schemes. That is, it is a scheme G with the equivalent properties* there is a group law expressible as a multiplication ? and inversion map ? on G; or...
. That is, for any integer we can consider the morphism on the multiplicative group that takes -th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism that serves as the identity.

The resulting group scheme is written '. It gives rise to a reduced scheme, when we take it over a field , if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
the characteristic of does not divide . This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent elements in their structure sheaves); for example ' over a finite field
Finite field

In abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory....
with elements for any prime number
Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
.

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 (theory of Pierre Cartier
Pierre Cartier (mathematician)

Pierre Cartier is a mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory....
). The Galois cohomology of this group scheme is a way of expressing Kummer theory
Kummer theory

In mathematics, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field....
.

See also

multiplicative group of integers modulo n
Multiplicative group of integers modulo n

In modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n....
additive group
Additive group

In mathematics, an additive group may be*an abelian group, when it is written using the symbol + for its binary operation*the underlying group under addition of a field , ring , vector space or other structure having addition as one of its operations...