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 groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...

, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible 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 of ''F'' and the

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

• 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 In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in Projective scheme, projective algebraic geometry and toric ...

GL(1)..

Examples

*The multiplicative group of integers modulo ''n'' is the group under multiplication of the invertible elements of

\mathbb/n\mathbb

. When ''n'' is not prime, there are elements other than zero that are not invertible. * The multiplicative group of

positive 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 f ...

\mathbb^+

is an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

with 1 its identity element. 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 ...

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 gr ...

of this group to the additive group of real numbers,

\mathbb

. * The multiplicative group of a field

F

is the set of all nonzero elements:

F^\times = F -\

, under the multiplication operation. If

F

is finite of order ''q'' (for example ''q'' = ''p'' a prime, and

F = \mathbb F_p=\mathbb Z/p\mathbb Z

), then the multiplicative group is cyclic:

F^\times \cong C_

Group scheme of roots of unity

The group scheme of ''n''-th roots of unity is by definition the kernel of the ''n''-power map on the multiplicative group GL(1), considered as a group scheme. 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, with the morphism ''e'' that serves as the identity. The resulting group scheme is written μ_''n'' (or

\mu\!\!\mu_n

). It gives rise to a reduced scheme, 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 bic ...

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 μ_''p'' over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

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 (mathematics), 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 ...

''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.

Notes

References

* Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004.

Examples

Group scheme of roots of unity

Notes

References

See also