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Outside of number theory the simpler notation is often used, though it can be confused with the p-adic integers when n is a prime number. The multiplicative group of integers modulo n, which is the group of units in this ring, may be written as (depending on the author) (/), (/), (/), (/) (for German Einheit, which translates as unit), , or ...
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.
A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch( G ) of these morphisms forms an abelian group under pointwise multiplication.
The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if x m = 1 and y n = 1, then (x −1) m = 1, and (xy) k = 1, where k is the least common multiple of m and n. Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.
In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication.
The circle group is more than just an abstract algebraic object. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on , the circle group has the structure of a topological group.
The 2-adic integers, with selected corresponding characters on their Pontryagin dual group. In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and ...
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
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