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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).
Since multiplication and inversion are continuous functions on , the circle group has the structure of a topological group. Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of C × {\displaystyle \mathbb {C} ^{\times }} (itself regarded as a topological group).
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix ...
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.
The multiplicative group of the field is the group whose underlying set is the set of nonzero real numbers {} and whose operation is multiplication. More generally, one speaks of an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted 0 {\displaystyle 0} , and the inverse of ...
the value group or valuation group Γ v = v(K ×), a subgroup of Γ (though v is usually surjective so that Γ v = Γ); the valuation ring R v is the set of a ∈ K with v(a) ≥ 0, the prime ideal m v is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of R v), the residue field k v = R v /m v,
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.
The S-unit equation is a Diophantine equation. u + v = 1. with u and v restricted to being S-units of K (or more generally, elements of a finitely generated subgroup of the multiplicative group of any field of characteristic zero).