Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integral Basis for quartic Kummer extensions over $\mathbb{Z}[ι]$

S. Venkataraman, Manisha V. Kulkarni

Abstract

Let $K=\mathbb{Q}[ι]$ and $N=K[\sqrt[4]α]$, $α\in\mathbb{Z}[ι]$, $alpha=fg^2h^3$, $f$, $g$, $h\in \mathbb{Z}[ι]$ are pairwise coprime and square free. Let $\mathcal{O}_N$ be the ring of integers of $N$. In this article we construct normalised integral basis for $\mathcal{O}_N$ over $\mathbb{Z}[ι]$, that is an integral basis of the form \[ \left\{1,\frac{f_1(α)}{d_1},\frac{f_2(α)}{d_2},\frac{f_{3}(α)}{d_3}\right\} \] where $d_i \in \mathbb{Z}[i]$ and $f_i(X)$, $\leq i\leq 3$ are monic polynomials of degree $i$ over $\mathbb{Z}[ι]$. We explicitly determine what $d_i$, $1\leq i\leq n-1$ are in terms of $f$, $g$ and $h$.

Integral Basis for quartic Kummer extensions over $\mathbb{Z}[ι]$

Abstract

Let and , , , , , are pairwise coprime and square free. Let be the ring of integers of . In this article we construct normalised integral basis for over , that is an integral basis of the form where and , are monic polynomials of degree over . We explicitly determine what , are in terms of , and .

Paper Structure

This paper contains 6 sections, 17 theorems, 76 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic Sub Extension
  4. Ramification Groups
  5. Integral Basis
  6. Appendix

Key Result

theorem 1

There exist $d_{1},d_{2},\ldots,d_{n}\in{\cal O}_{ F}$ and monic polynomials $f_{i}(X)\in {\cal O}_{ F}[X]$, $1\leq i\leq n-1$, $deg(f_{i}(X))=i$, such that is a basis for ${\cal O}_{ L}$ over ${\cal O}_{ F}$. Further, $d_{i}$'s satisfy the following conditions:

Theorems & Definitions (27)

  • theorem 1: Normalised Integral Basis
  • lemma 1
  • proposition 1
  • proof
  • definition 1
  • proposition 2
  • proposition 3
  • definition 2
  • proposition 4
  • proposition 5
  • ...and 17 more