Narrow Results

A well known result of Schur states that if n is a positive integer and a₀, a₁,…,a_n are arbitrary integers with a₀a_n coprime to n!, then the polynomial is irreducible over the field ℚ of rational numbers. In case each a_i = 1, it is known that the Galois group of f_n(x) over ℚ contains A_n, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials f_n(x) which leads to the construction of trinomials of degree n for each n with Galois group S_n, the symmetric group on n letters.

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on $\lambda$-invariants in a $p$-extension of ${\mathbb{Z}}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched ${\mathbb{Z}}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

A set of real nth roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory.

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra $R^{\underline{coA}}$ with a certain subalgebra of the smash product $R\#Â$. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.

For f(x) ∈ ℤ[x] and a ∈ ℤ, we let fⁿ(x) be the nth iterate of f(x), P(f, a) = {p prime: p|fⁿ(a) for some n}, and D(P(f, a)) denote the natural density of P(f, a) within the set of primes. A conjecture of Jones [5] indicates that D(P(f, a)) = 0 for most quadratic f. In this paper, we find an exceptional family of (f, a) such that D(P(f, a)) > 0 by considering f_t(x) = (x + t)² - 2 - t and a_t = f_t(0) for t ∈ ℤ. We prove that if t is not of the form ±M² ± 2 or ±2M² ± 2, then D(P(f_t, a_t)) = ⅓. We also determine D(P(f_t, a_t)) in some cases when the density is not equal to ⅓. Our results suggest a connection between the arithmetic dynamics of the conjugates of x² and the conjugates of x² - 2.

In this article, we investigate systems of fundamental units of quartic fields which are constructed by adjoining a root of a certain parametric quartic polynomial to ℚ.

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions $1+\mathrm{\text{log}}(1-x)$ and $\mathrm{\text{cos}}x$. We first show that the Galois groups of the $n$th Taylor polynomials of $1+\mathrm{\text{log}}(1-x)$ are as large as possible, namely, $S_n$ (full symmetric group) or $A_n$ (alternating group), depending on the residue of the integer number $n$ modulo $4$. We then compute the Galois groups of the $n$th Taylor polynomials of $\mathrm{\text{cos}}(x)$ and show that these Galois groups essentially coincide with the Coexter groups of type $B_n$ (or an index 2 subgroup of the corresponding Coexter group).

In this paper, we study the symmetries of (a particular kind of) coherent states defined in the framework of the Galois quantum theory introduced in a previous publication. The configuration and the momentum spaces of this theory are given by finite and discrete abelian groups, namely the Galois group G = Gal(L : K) of a Galois field extension (L : K) and its unitary dual Ĝ ≐ Hom_gr (G, U(1)). The main interest of this quantum theory is that it is possible to define coherent states with indeterminacies in the position q ∊ G and the momentum χ ∊ Ĝ encoded by subgroups of G and Ĝ respectively. First, we show that the group of automorphisms of a coherent state with indeterminacy H ⊆ G in the position is H × H^⊥, where H^⊥ is the annihilator of H in Ĝ and encodes the corresponding indeterminacy in the momentum. Second, we show that the quantum numbers that completely define such coherent states fix an irreducible unitary representation of H × H^⊥. These results generalize the group-theoretical interpretation of the limit cases of Heisenberg indeterminacy principle proposed in previous publications to states with non-zero indeterminacies in both q and p. According to this interpretation, a quantum coherent state describes a structure-endowed system characterized by a group of automorphisms acting in an irreducible unitary representation fixed by the quantum numbers that define the state.