Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the monogenity of quartic number fields defined by $x^4+ax^2+b$

Lhoussain El Fadil, István Gaál

Abstract

For any quartic number field $K$ generated by a root $α$ of an irreducible trinomial of type $x^4+ax^2+b\in Z[x]$, we characterize when $Z[α]$ is integrally closed. Also for $p=2,3$, we explicitly give the highest power of $p$ dividing $i(K)$, the common index divisor of $K$. For a wide class of monogenic trinomials of this type we prove that up to equivalence there is only one generator of power integral bases in $K=Q(α)$. We illustrate our statements with a series of examples.

On the monogenity of quartic number fields defined by $x^4+ax^2+b$

Abstract

For any quartic number field generated by a root of an irreducible trinomial of type , we characterize when is integrally closed. Also for , we explicitly give the highest power of dividing , the common index divisor of . For a wide class of monogenic trinomials of this type we prove that up to equivalence there is only one generator of power integral bases in . We illustrate our statements with a series of examples.
Paper Structure (15 sections, 7 theorems, 47 equations)

This paper contains 15 sections, 7 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Monogenity of number fields and polynomials
  3. Results on the monogenity of binomilas, trinomials
  4. The index of a number field
  5. The purpose of the present paper
  6. Main Results
  7. Integral closedness of $\mathbb Z[\alpha]$
  8. The divisibility of the field index by $2$ and $3$
  9. The number of inequivalent generators of power integral bases
  10. A short introduction to Newton polygons
  11. Examples
  12. Examples 1. Some monogenic number fields defined by non-monogenic trinomials
  13. Examples 2: Monogenic and non-monogenic fields, index forms.
  14. Examples 3: Applying Engstrom's results.
  15. Proofs of our main results

Key Result

Theorem 2.1

The ring $\mathbb Z[\alpha]$ is the ring of integers of $K$ if and only if for every prime integer $p$, $p$ satisfies one of the following conditions: In particular, if for every prime integer $p$, $p$ satisfies one of these conditions, then $i(K)$ is trivial, that is $i(K)=1$.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 1
  • Theorem 2.4
  • Theorem 3.1
  • Remark 2
  • Proposition 4.1
  • Lemma 5.1