Suppose K is a local field with finite residue field of characteristic p 6= 2 and K<p(M) is its maximal p-extension such that Gal(K<p(M)/K) has period p M and nilpotent class < p. If charK = 0 we assume that K contains a primitive p -th root of unity. The paper contains an overview of methods and results describing the structure of this Galois group together with its filtration by ramification ...

Let Cl(OK [G]) denote the locally free class group, that is the group of stable isomorphism classes of locally free OK [G]-modules, where OK is the ring of algebraic integers in the number field K and G is a finite group. We show how to compute the Swan subgroup, T (OK [G]), of Cl(OK [G]) when K = Q(ζp), ζp a primitive p-th root of unity, G = C2, where p is an odd (rational) prime so that hp = ...

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F . We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F . For almost all completions of the theory, we show that algebraic closure agrees with definable closure, as soon as A contains the relative algebraic closure of the prime field.

Let α be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates αi lie in a sector | arg z| ≤ θ. In 1995, G. Rhin and C. Smyth computed the greatest lower bound c(θ) of the absolute Mahler measure ( ∏d i=1 max(1, |αi|)) of α, for θ belonging to nine subintervals of [0, 2π/3]. More recently, in 2004, G. Rhin and Q. Wu improved the result to thirteen subintervals of [...

Let μ and ν = (ν1, . . . , νk) be partitions such that μ is obtained from ν by adding m parts of size r. Descouens and Morita proved algebraically that the modified Macdonald polynomials H̃μ(X; q, t) satisfy the identity H̃μ = H̃νH̃(rm) when the parameter t is specialized to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ νk and r ∈ {1, 2}. This note giv...

In this paper we first introduce a new generalized cyclotomy of order 2 with respect to pe1 1 2pet t , then we calculate the new cyclotomic numbers of order 2. Some applications of the new cyclotomy in sequences, cryptography, and coding theory are also discussed. In the last section of this paper, we introduce more generalized cyclotomies and point out their applications. The major motivation ...

When G has size n and g ∈ G, for any character χ of G we have χ(g)n = χ(gn) = χ(1) = 1, so the values of χ lie among the nth roots of unity in S1. More precisely, the order of χ(g) divides the order of g (which divides #G). Characters on finite abelian groups were first studied in number theory, since number theory is a source of many interesting finite abelian groups. For instance, Dirichlet u...

The existence is shown of a set of (p~ -1) generalized Hadamard matrices H(p, p~'~) of order p2'~, each of which is symmetric and regular. When normalized to become unitary matrices, they form a multiplicative group of order p'~, simply isomorphic to the additive group of GF(pm). The rows of these (p~ 1) matrices are shown to be the image, under the well-known isomorphic mapping relating the pt...

In [LLT1], Lascoux, Leclerc and Thibon give some factorisation formulas for Hall-Littlewood functions when the parameter q is specialized at roots of unity. They also give formulas in terms of cyclic characters of the symmetric group. In this article, we give a generalization of these specializations for different versions of the Macdonald polynomials and we obtain similar formulas in terms of ...

Let p be an odd prime and ζ be a primitive p-root of unity. For any integer a prime to p, let ( p ) denote the Legendre symbol, which is 1 if a is a square mod p, and is −1 otherwise. Using Euler’s Criterion that a(p−1)/2 = ( p ) mod p, it follows that the Legendre symbol gives a homomorphism from the multiplicative group of nonzero elements Fp of Fp = Z/pZ to {±1}. Gauss’s law of quadratic rec...