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Weisstein, Eric W. "Modulo Multiplication Group". MathWorld. Weisstein, Eric W. "Primitive Root". MathWorld. Web-based tool to interactively compute group tables by John Jones; OEIS sequence A033948 (Numbers that have a primitive root (the multiplicative group modulo n is cyclic))
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.
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.
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 ...
The Conway polynomial is chosen to be primitive, so that each of its roots generates the multiplicative group of the associated finite field. The field F p n contains a unique subfield isomorphic to F p m for each m dividing n, and this accounts for all the subfields of F p n. For any m dividing n the cyclic group F * p n contains a subgroup ...
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 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.
In the case that m is a prime, say p, then () = and all the non-zero elements of / have multiplicative inverses, thus / is a finite field. In this case, the multiplicative group of integers modulo p form a cyclic group of order p − 1.