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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 multiplicative group of -adic units is a product of the finite group of roots of unity and a group isomorphic to the -adic integers. The finite group is cyclic of order p − 1 {\displaystyle p-1} or 2 {\displaystyle 2} , as p {\displaystyle p} is odd or even, respectively, and so it is isomorphic to ( Z / q Z ) × {\displaystyle (\mathbb ...
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 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 ...