On automorphism groups of cyclotomic function fields over finite fields

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme over a finite near-field K. It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V ), where V is the linear space associated with K. A sufficient condition on a cyclotomic scheme C that guarantee the inclusion Aut(C)≤ A L(1,F), where F is a finite field with |K| elements, is given.

Automorphism groups of hyperelliptic function fields

The Automorphism Group over Finite Fields

It is shown that the invertible polynomial maps over a finite field Fq, if looked at as bijections Fq −→ Fq , give all possible bijections in case q = 2, or q = p where p > 2. In case q = 2 where r > 1 it is shown that the tame subgroup of the invertible polynomial maps give only the even bijections, i.e. only half the bijections. As a consequence it is shown that a set S ⊂ Fq can be a zero set...

On Galois Groups of Abelian Extensions over Maximal Cyclotomic Fields

Let k0 be a finite algebraic number field in a fixed algebraic closure Ω and ζn denote a primitive n-th root of unity ( n ≥ 1). Let k∞ be the maximal cyclotomic extension of k0, i.e. the field obtained by adjoining to k0 all ζn ( n = 1, 2, ...). Let M and L be the maximal abelian extension of k∞ and the maximal unramified abelian extension of k∞ respectively. The Galois groups Gal(M/k∞) and Gal...

On the Reducibility of Cyclotomic Polynomials over Finite Fields

The irreducibility of a polynomial in a prime modulus implies irreducibility over the rationals. However, the converse is certainly not true. In fact, there are some polynomials that are irreducible over the rationals yet reducible in every prime modulus. We show that the nth cyclotomic polynomial reduces modulo all primes if and only if the discriminant of the nth cyclotomic polynomial is a sq...