The main goal of this paper is to calculate the index of any number field K generated by a root of an irreducible trinomial x4+ax+b∈Zx. Our approach is based on Engstrom’s results and the ...factorization of 2ZK and 3ZK. In particular, we reformulate the Davis and Spearman’s results. Namely, for every prime integer p, we evaluate νp(i(K)). The existence of a common index divisor of K guaranties the non monogenity of K.
We investigate the Vorono\"{\i} summation problem for ${\rm GL}_n$ in the level aspect for $n\geq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified, a ...common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give classical refinements of our formula and study the $p$-adic generalisations of the Bessel transform.
In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial , we describe all prime power divisors of the index of K answering Problem 22 of Narkiewicz 26. In ...particular, if , then K is not mongenic. We illustrate our results by some computational examples.
Let be a number field generated by a complex root α of a monic irreducible trinomial In this paper, for every prime integer p, we give necessary and sufficient conditions on a and b so that p is a ...common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic.
For a number field K defined by a trinomial F (x) = x6 + ax + b ∈ ℤx, Jakhar and Kumar gave some necessary conditions on a and b, which guarantee the non-monogenity of K 25. In this paper, for every ...prime integer p, we characterize when p is a common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic. In such a way our proposed results extend those of Jakhar and Kumar.
This article gives a nice family of rearrangements of a conditionally convergent series for use in calculus or first analysis classes. In each series, the positive terms and the negative terms both ...form a harmonic series. For each natural number k, a series is given that converges to the logarithm of k. The series are wonderful to show to any calculus class even if the instructor omits the details. The series or variations are known.
We use the theta correspondence to study the equivalence between Godement–Jacquet and Jacquet–Langlands L-functions for
${\mathrm {GL}}(2)$
. We show that the resulting comparison is in fact an ...expression of an exotic symmetric monoidal structure on the category of
${\mathrm {GL}}(2)$
-modules. Moreover, this enables us to construct an abelian category of abstractly automorphic representations, whose irreducible objects are the usual automorphic representations. We speculate that this category is a natural setting for the study of automorphic phenomena for
${\mathrm {GL}}(2)$
, and demonstrate its basic properties. This paper is a part of the author’s thesis 4.
The notion of the neutrosophic triplet was introduced by Smarandache and Ali. This notion is based on the fundamental law of neutrosophy that for an idea X, we have neutral of X denoted as neut(X) ...and anti of X denoted as anti(X). This paper studied a neutrosophic triplet set which is a collection of all triple of three elements that satisfy certain properties with some binary operation. Also given some interesting properties related to them. Further, in this paper investigated that from the neutrosophic triplet group can construct a classical group under multiplicative operation for ℤn, for some specific n. These neutrosophic triplet groups are built using only modulo integer 2p, with p is an odd prime or Cayley table.
PDF
