Leírás
Online Number Theory Seminar
Abstract: Let $K = \mathbb{Q}(\theta)$ be a number field of degree $n$ with $\theta$ a root of a monic irreducible polynomial $F(x)$ in $\mathbb{Z}[x]$, and $\mathbb{Z}_{K}$ its ring of integers. The field $K$ is monogenic if $\mathbb{Z}_K$ admits a power integral basis of the form $(1,\eta,\ldots,\eta^{n-1})$ for some primitive element $\eta\in \mathbb{Z}_K$, that is $\mathbb{Z}_K=\mathbb{Z}[\eta]$. The fundamental method to test whether $K$ is monogenic or not and determine all the power integral bases is to solve an index form equation $I(x_2, \ldots, x_n) = \pm 1$ relative to an integral basis of $K$. For index form equations, there are general effective finiteness results due to Gy\H{o}ry, and efficient algorithms for several classes of number fields, mainly those given by Ga\'{a}l, Gy\H{o}ry, Peth\H{o}, Pohst and their collaborators. Recently, many authors based their approach on prime ideal factorization via Newton polygon techniques as introduced by Ore, and developed by Gu\`{a}rdia, Montes and Nart. They studied the problem of the monogenity and indices in various families of number fields. In my talk, I will begin by recalling some fundamental results regarding monogenity, not monogenity, and indices in number fields. Following that, I will present some definitions and results concerning the application of Newton polygon techniques in the decomposition of primes in number fields. Then I will provide an overview of my recent results on this topic, focusing on some infinite parametric families of pure number fields $\mathbb{Q}(\sqrt[n]{m})$, and number fields defined by irreducible trinomials of the type $x^n+ax^m+b$. This talk is partly based on joint works with Boudine, Didi, El Fadil and Teibekabe.
For access please contact the organizers (ntrg[at]science.unideb.hu).