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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 (F ∖ {0}, •), where 0 refers to the zero element of F and the binary operation • is the field multiplication, the algebraic torus GL(1).
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element if it is a primitive (q − 1) th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as α i for some natural number i.
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, subtraction and division are defined and satisfy certain basic rules.
Let V be one-dimensional. Then any finite group faithfully acting on V is a subgroup of the multiplicative group of the field K, and hence a cyclic group.It follows that G consists of roots of unity of order dividing n, where n is its order, so G is generated by pseudoreflections.
The map sending a finite-dimensional F-vector space to its dimension induces an isomorphism for any field F. Next, =, the multiplicative group of F. [1] The second K-group of a field is described in terms of generators and relations by Matsumoto's theorem.
The multiplicative group of integers modulo n, which is the group of units in this ring, ... By the fundamental theorem of finite abelian groups, the group ...
The group of units, R ×, can be decomposed as a direct product G 1 ×G 2, as follows. The subgroup G 1 is the group of (p r – 1)-th roots of unity. It is a cyclic group of order p r – 1. The subgroup G 2 is 1+pR, consisting of all elements congruent to 1 modulo p. It is a group of order p r(n−1), with the following structure:
Then the exact sequence of group cohomology shows that there is an isomorphism between A G /π(A G) and Hom(G,C). Kummer theory is the special case of this when A is the multiplicative group of the separable closure of a field k, G is the Galois group, π is the nth power map, and C the group of nth roots of unity.