Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1002.0158

Simplest Cubic Fields

Q. Mushtaq·S. Iqbal
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $Q(α)$ be the simplest cubic field, it is known that $Q(α)$ can be generated by adjoining a root of the irreducible equation $x^{3}-kx^{2}+(k-3)x+1=0$, where $k$ belongs to $Q$. In this paper we have established a relationship between $α$, $α'$ and $k,k'$ where $α$ is a root of the equation $x^{3}-kx^{2}+(k-3)x+1=0$ and $α'$ is a root of the same equation with $k$ replaced by $k'$ and $Q(α)=Q(α')$.

Explore related subjects

math.NTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Q. Mushtaq, S. Iqbal. 2010-01-31. Simplest Cubic Fields. https://arxiv.org/abs/1002.0158

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗