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Integers for Simple Radical Extensions

Julius Kraemer

TL;DR

This work provides an explicit, modular approach to determining the ring of integers and discriminant for number fields generated by simple radical extensions $K=\mathbb{Q}(\alpha)$ with $\alpha^n=a$. By decomposing $n$ into prime-power factors and exploiting linear disjointness, the authors reduce the problem to prime-power cases and construct suitable $q'$-maximal and $p$-maximal orders, together with precise $p$-radicals and integral bases. They derive concrete discriminant formulas, including $\Delta_K=\frac{\prod_j \Delta_j^{n_j}}{a^d}$, with $d$ depending on the decomposition and Wieferich-type valuations, and provide a robust framework for assembling the global ring of integers from local maximal orders via Dedekind-radical theory. The results extend to non-coprime cases ($p\mid a$) through a reduction to a related prime-power extension and cover all primes, including $p=2$, with careful explicit constructions and proofs of maximality. Overall, the paper unifies and extends prior work on integral bases and discriminants for pure extensions, offering actionable, formula-driven methods for arithmetic in simple radical fields.

Abstract

The ring of integers and the discriminant are determined for number fields which are simple radical extensions.

Integers for Simple Radical Extensions

TL;DR

This work provides an explicit, modular approach to determining the ring of integers and discriminant for number fields generated by simple radical extensions with . By decomposing into prime-power factors and exploiting linear disjointness, the authors reduce the problem to prime-power cases and construct suitable -maximal and -maximal orders, together with precise -radicals and integral bases. They derive concrete discriminant formulas, including , with depending on the decomposition and Wieferich-type valuations, and provide a robust framework for assembling the global ring of integers from local maximal orders via Dedekind-radical theory. The results extend to non-coprime cases () through a reduction to a related prime-power extension and cover all primes, including , with careful explicit constructions and proofs of maximality. Overall, the paper unifies and extends prior work on integral bases and discriminants for pure extensions, offering actionable, formula-driven methods for arithmetic in simple radical fields.

Abstract

The ring of integers and the discriminant are determined for number fields which are simple radical extensions.

Paper Structure

This paper contains 15 sections, 49 theorems, 33 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Prelimininaries
  4. Preliminaries for Simple Radical Extensions
  5. A $q$-maximal Order for $q$ not Equal to $p$
  6. A $p$-maximal Order in the Coprime Case
  7. A $p$-maximal Order in the Non-Coprime Case
  8. Determination of a p-maximal Order $\mathcal{O}_p$ and the Discriminant
  9. Determination of the p-radical $I_p$ of $\mathcal{O}_p$
  10. Proof of the p-maximality of $\mathcal{O}_p$
  11. Ring of Integers for Simple Radical Extensions
  12. Integers as Product of $q'$-maximal Orders
  13. Linearly Disjoint Extensions
  14. q-Radicals for Dedekind Rings
  15. Integers for Simple Radical Extensions

Key Result

Lemma 2.3

Suppose $L = \mathbb{Q}(\lambda)$ a number field of degree $n$ with ring of integers $\mathcal{O}_L$ and orders $\mathcal{O}'$ and $\mathcal{O}"$ with $\mathcal{O}' \subset \mathcal{O}"$. Suppose $B' = \{\varrho_i \;|\; 1 \le i \le n\}$ and $B"$ are integral bases of $\mathcal{O}'$ and $\mathcal{O}"

Theorems & Definitions (135)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 125 more