Corpus ID: 119682374

Binomial Thue equations and power integral bases in pure quartic fields

@article{Gaal2018BinomialTE,
  title={Binomial Thue equations and power integral bases in pure quartic fields},
  author={Istv'an Ga'al and L{\'a}szl{\'o} Remete},
  journal={arXiv: Number Theory},
  year={2018}
}
It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem is especially delicate to consider in an infinite parametric family of number fields. In the present paper we investigate power integral bases in the infinite parametric family of pure quartic fields $K=Q(\sqrt[4]{m})$. We often pointed out close connection of various types of Thue equations with calculating… Expand
Integral bases and monogenity of pure fields
Abstract Let m be a square-free integer ( m ≠ 0 , ± 1 ). We show that the structure of the integral bases of the fields K = Q ( m n ) is periodic in m. For 3 ≤ n ≤ 9 we show that the period length isExpand
On power integral bases of certain pure number fields defined by $x^{2^u\cdot3^v}-m$
Let K = Q(α) be a pure number field generated by a complex root α of a monic irreducible polynomial F(x) = x ·3 − m, with m , ±1 a square free rational integer, u, and v two positive integers. InExpand
On power integral bases of certain pure number fields defined by $$x^{42} - m$$
Let K be a pure number field generated by a complex root of a monic irreducible polynomial F(x) = x − m ∈ Z[x], with m , ±1 a square free integer. In this paper, we study the monogeneity of K. WeExpand
On power integral bases of certain pure number fields defined by $$x^{3^r} - m$$
Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{3^r} -m$$ , with $$m \ne \pm 1$$ is a square-free rationalExpand
Non-Monogenity of an Infinite Family of Pure Octic Fields
Let m be a square free integer. The aim of this paper is to prove that the infinite family of pure octic field L Q √m is non-monogenic if m mod , ultimately, to complete the classification of pureExpand
On power integral bases for certain pure number fields defined by
Abstract Let be a pure number field generated by a root α of a monic irreducible polynomial with is a square free integer, r and s are two positive integers. In this article, we study the monogenityExpand
Two Families of Monogenic $S_4$ Quartic Number Fields
Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, andExpand
The Monogeneity of Kummer Extensions and Radical Extensions.
We give necessary and sufficient conditions for the Kummer extension $\mathbb{Q}\left(\zeta_n,\sqrt[n]{\alpha}\right)$ to be monogenic over $\mathbb{Q}(\zeta_n)$ with $\sqrt[n]{\alpha}$ as aExpand
On monogenity of certain pure number fields defined by $$x^{20}-m$$
  • L. E. Fadil
  • Mathematics
  • São Paulo Journal of Mathematical Sciences
  • 2021
Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a complex root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{20}-m$$ , with $$m \ne \mp 1$$ a square free rationalExpand
A note ON MONOGENEITY of pure number fields
Gassert’s paper ”A NOTE ON THE MONOGENEITY OF POWER MAPS” is cited at least by 17 papers in the context of monogeneity of pure number fields despite some errors that it contains and remarks on it. InExpand
...
1
2
...

References

SHOWING 1-10 OF 14 REFERENCES
Power integral bases in parametric families of biquadratic fields
We consider two families of totally complex biquadratic fields depending on two parameters. These families were recently considered by J.G.Huard, B.K.Spearman and K.S.Williams [8]. Using our generalExpand
Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields
LetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutionsExpand
Binomial Thue equations, ternary equations and power values of polynomials
AbstractWe explicitly solve the equation Axn  − Byn = ±1 and, along the way, we obtain new results for a collection of equations Axn  − Byn = zm with m ∈ {3, n}, where x, y, z, A, B, and n areExpand
Power integral bases in orders of families of quartic fields
We consider five infinite families of polynomials, namely I f1(x) = x4 + k (k > 0, k 6= 4k4 0) II f2(x) = x4 + k2x2 − 2kx + 1 III f3(x) = x4 + x3 + kx2 + εx + 1 (k > 0, ε = ±1, k 6= 2 if ε = 1) IVExpand
Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1
Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional ``hypergeometric method'' for rational and algebraic approximationExpand
Relative power integral bases in infinite families of quartic extensions of quadratic field
We consider infinite parametric families of octic fields, that are quartic extensions of quadratic fields. We describe all relative power integral bases of the octic fields over the quadraticExpand
On the Resolution of Thue Inequalities
  • A. Pethö
  • Mathematics, Computer Science
  • J. Symb. Comput.
  • 1987
Let F(x, y) @? Z[x, y] be a homogenous polynomial of degree at least 3, and m @? Z. We describe a method for the resolution in (x, y) @? Z^2, |y| =< y"0 of the inequality |F(x, y)| =< m, using theExpand
On the existence of canonical number system in certain classes of pure algebraic number fields
  • J. Prime Research in Math
  • 2011
...
1
2
...