Let $K=\mathbb Q(θ)$ be an algebraic number field with $θ$ a root of an irreducible trinomial $f(x)=x^6+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, for each prime number $p$ we compute the highest power of $p$ dividing the discriminant of $K$ in terms of the prime powers dividing $a,~b$ and discriminant of $f(x)$. An explicit $p$-integral basis of $K$ is also given for each prime $p$ and a method is described to obtain an integral basis of $K$ from these $p$-integral bases which is illustrated with examples.
Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-α^2)(x^2-(α+1)^2).$ We express their integer solutions by means of the divisors of the discriminant of $f(x),$ where $y^2=f(x)$.
Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $α$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \] The totally real cyclic fields $K=Q(α)$ are called simplest sextic fields and are well known in the literature. Using a completely new approach we explicitly give an integral basis of $K$ in a parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$ in which cases we give all generators of power integral bases.
Let $K = \Q(θ)$ be an algebraic number field with $θ$ satisfying an irreducible polynomial $x^{9} - a$ over the field $\Q$ of rationals and $\Z_K$ denote the ring of algebraic integers of $K$. In this article, we provide the exact power of each prime which divides the index of the subgroup $\Z[θ]$ in $\Z_K$. Further, we give a $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to a construction of an integral basis of $K$ which is illustrated with examples.
We adapt an old local-to-global technique of Ore to compute, under certain mild assumptions, an integral basis of a number field without a previous factorization of the discriminant of the defining polynomial. In a first phase, the method yields as a by-product successive splittings of the discriminant. When this phase concludes, it requires a squarefree factorization of some base factors of the discriminant to terminate.
We determine an explicit triangular integral basis for any separable cubic extension of a rational function field over a finite field in any characteristic. We obtain a formula for the discriminant of every such extension in terms of a standard form in a tower for the Galois closure.
We show that the Weil representation associated with any discriminant form admits a basis in which the action of the representation involves algebraic integers. The action of a general element of $\operatorname{SL}_{2}(\mathbb{Z})$ on many parts of these bases is simple an explicit, a fact that we use for determining the dimension of the space of invariants for some families of discriminant forms.
We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, o…
In this paper we improve the upper bound of the number $N_{K, n}(X)$ of degree $n$ extensions of a number field $K$ with absolute discriminant bounded by $X$. This is achieved by giving a short $\mathcal{O}_K$-basis of an order of an extension $L$ of $K$. Our result generalizes the best known upper bound on $N_{\mathbb{Q}, n}(X)$ by Lemke Oliver and Thorne to all number fields $K$. Precisely, we prove that $N_{K, n}(X) \ll_{K, n} X^{c (\log n)^2}$ for an explicit constant $c$ independent on $K$ and $n$. We also improve the upper bound of the number of maximal arithmetic subgroups in certain co…
Let $K$ be a number field, and $S$ a finite set of places in $K$ containing all infinite places. We present an implementation for solving the $S$-unit equation $x + y = 1$, $x,y \in\mathscr{O}_{K,S}^\times$ in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets $S$. As an application, we prove an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded disc…
If a number field has a large degree and discriminant, the computation of the class number becomes quite difficult, especially without the assumption of GRH. In this article, we will unconditionally show that a certain nonabelian number field of degree 120 has class number one. This field is the unique $A_5 \times C_2$ extension of the rationals that is ramified only at 653 with ramification index 2. It is the largest degree number field unconditionally proven to have class number 1. The proof uses the algorithm of Guàrdia, Montes and Nart to calculate an integral basis and then finds integral…
In this work, we use the notion of ``symmetry'' of functions for an extension $K/L$ of finite fields to produce extensions of a function field $F/K$ in which almost all places of degree one split completely. Then we introduce the notion of ``quasi-symmetry'' of functions for $K/L$, and demonstrate its use in producing extensions of $F/K$ in which all places of degree one split completely. Using these techniques, we are able to restrict the ramification either to one chosen rational place, or entirely to non-rational places. We then apply these methods to the related problem of building asympto…
