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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.
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.
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 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:
In modular arithmetic, the integers coprime (relatively prime) to n from the set {,, …,} of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n.
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 inverse for an element a of a finite field can be calculated a number of different ways: By multiplying a by every number in the field until the product is one. This is a brute-force search. Since the nonzero elements of GF(p n) form a finite group with respect to multiplication, a p n −1 = 1 (for a ≠ 0), thus the inverse ...
Suppose that G is a group scheme of finite type over a field k. Let G 0 be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then G is an extension of a finite étale group scheme by G 0. G has a unique maximal reduced subscheme G red, and if k is perfect, then G red is a smooth group variety that is a ...