Related papers: Some split symbol algebras of prime degree

Let $\mathcal{O}$ be a maximal order in the quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$. We prove two theorems that allow us to recover the structure of $\mathcal{O}$ from limited information. The first says that for…

We look at connection Laplacians L,g defined by a field h:G to K, where G is a finite set of sets and K is a normed division ring which does not need to be commutative, nor associative but has a conjugation leading to the norm as the square…

Let K/k be an Abelian extension of number fields, S be a set of places of k, and p be an odd prime number. We continue an earlier investigation of the author into the values at zero of the S-imprimitive partial zeta functions of K/k. An…

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(\psi:\,f\mapsto V_p(f)\) from the divisor…

Let a Lie algebra $\mathfrak q$ be a linear sum of two complementary subalgebras $\mathfrak h$ and $\mathfrak r$. We continue our investigations initiated in (J. London Math. Soc. 103 (2021), 1577-1595), where compatible Poisson brackets…

We derive the initial condition for the perturbative fragmentation function of a heavy quark through order O(alpha_s^2) in the MS-bar scheme. This initial condition is useful for computing heavy quark (or lepton, in case of QED) energy…

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is…

To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely…

We construct the integrals of motion for the 5D deformed Kepler system with non-central potentials in $su(2)$ Yang-Coulomb monopole field. We show that these integrals form a higher rank quadratic algebra $Q(3; L^{so(4)}, T^{su(2)})\oplus…

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial…

We determine the structure of the obstruction group of the Hasse norm principle for a finite separable extension $K/k$ of a global field of degree $d$, where $d$ has a square-free prime factor $p$ and a $p$-Sylow subgroup of the Galois…

In this paper we will study some properties of the matrix representations of symbol algebras of degree three, we study some equations with coefficients in these algebras, we find an octonion algebra in a symbol algebra of degree three, we…

Let $p$ be a prime and let $K$ be a finite extension of the field ${\bf Q}_p$ of $p$-adic numbers such that the group ${}_pK^\times$ has order $p$. The ${\bf F}_p$-space $K^\times\!/K^{\times p}$ carries a natural filtration coming from the…

By using the Calkin-Wilf tree, we prove the irrationality of numbers of the form $\alpha=\frac{\sqrt{N}+p}{q}$ where $N$ is a positive integer which is not a perfect square, $p$ is a rational integer such that $p^2<N$ and $q$ is a positive…

Let $p$ be a prime and let $S_2(\Gamma(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $\Gamma(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(\Gamma(p))$ in a natural way. Around 1928, Hecke proved…

Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for…

Following work of V\'elez, we explicitly describe the splitting of the integral prime 2 in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. With previous work of the second…

The purpose of this note is to compare the properties of the symbolic pseudo-differential calculus on the Heisenberg and on the Engel groups; nilpotent Lie groups of 2-step and 3-step, respectively. Here we provide a preliminary analysis of…

Let $k$ be a number field and $O$ the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of $O$ and derive some density theorems on distributions of the discriminants by…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…