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Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$

Dylan Scofield, Hanson Smith

TL;DR

The paper provides an explicit $2$-adic prime-splitting description in the ring of integers of radical extensions ${\mathbb Q}(\sqrt[n]{a})$ for irreducible $x^n-a$, completing the odd-prime case and enabling a full characterization of common $N$-index divisors via Pleasants’ framework. It integrates the Montes algorithm with Vélez’s results to obtain concrete factorization patterns of $2\mathcal{O}_{K}$ across all cases, and derives criteria for CNID and CNID-forcing scenarios, leading to constructive methods for radical extensions with prescribed numbers of ring generators. The work also develops two new constructions: (i) radical fields that are non-monogenic yet have no common index divisors, and (ii) rings requiring $N$ generators for any $N>1$, accompanied by extensive examples. Collectively, these results deepen understanding of monogenicity, index theory, and the interplay between prime splitting and integral basis structure in radical extensions, with clear implications for arithmetic of number fields and explicit class field considerations.

Abstract

Following work of Vélez, we explicitly describe the splitting of the integral prime 2 in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. With previous work of the second author, this fully describes the splitting of any prime in $\mathbb{Q}(\sqrt[n]{a})$. Using this description, we classify common index divisors (the primes whose splitting prevents the existence of a power integral basis for the ring of integers). Using work of Pleasants, we extend this to describe common $N$-index divisors (primes that divide the index of any order generated over $\mathbb{Z}$ by $N$ elements). We also present two novel constructions of non-monogenic fields with no common index divisors as well as constructions of number rings requiring $N$ ring generators for any $N>1$. Examples are provided throughout.

Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$

TL;DR

The paper provides an explicit -adic prime-splitting description in the ring of integers of radical extensions for irreducible , completing the odd-prime case and enabling a full characterization of common -index divisors via Pleasants’ framework. It integrates the Montes algorithm with Vélez’s results to obtain concrete factorization patterns of across all cases, and derives criteria for CNID and CNID-forcing scenarios, leading to constructive methods for radical extensions with prescribed numbers of ring generators. The work also develops two new constructions: (i) radical fields that are non-monogenic yet have no common index divisors, and (ii) rings requiring generators for any , accompanied by extensive examples. Collectively, these results deepen understanding of monogenicity, index theory, and the interplay between prime splitting and integral basis structure in radical extensions, with clear implications for arithmetic of number fields and explicit class field considerations.

Abstract

Following work of Vélez, we explicitly describe the splitting of the integral prime 2 in the radical extension , where is an irreducible polynomial in . With previous work of the second author, this fully describes the splitting of any prime in . Using this description, we classify common index divisors (the primes whose splitting prevents the existence of a power integral basis for the ring of integers). Using work of Pleasants, we extend this to describe common -index divisors (primes that divide the index of any order generated over by elements). We also present two novel constructions of non-monogenic fields with no common index divisors as well as constructions of number rings requiring ring generators for any . Examples are provided throughout.
Paper Structure (11 sections, 13 theorems, 33 equations, 3 figures)

This paper contains 11 sections, 13 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction and Main Theorem
  2. Organization
  3. Previous work
  4. The Montes algorithm and a theorem of Ore
  5. Proof of the Main Theorem
  6. Auxiliary Lemmas and Remarks
  7. Common $N$-Index Divisors
  8. Examples of the Main Theorem
  9. Non-Monogenic Fields with No Common Index Divisors
  10. Pleasants' Exceptional Case
  11. A Construction Involving the Index Form

Key Result

Theorem 1.1

Suppose $x^n-a\in \mathbb{Z}[x]$ is irreducible. Changing variables, we assume that $a$ is $n^{\text{th}}$ power free. Let $2{\mathcal{O}}_{\mathbb{Q}(\space\sqrt[n]{a})}$ denote the ideal generated by $2$ in the ring of integers of $\mathbb{Q}(\space\sqrt[n]{a})$. The ideal factors of $2{\mathcal{O Using indices to indicate the relevant factorization, $1\leq i\leq r$ corresponds to a factorizatio

Figures (3)

  • Figure 1: Roadmap for the proof of Cases \ref{['MainIII']} and \ref{['MainIV']}
  • Figure 2: Possibilities for the left-most side as $w_0$ varies
  • Figure 3: The 9-sided principal $x$-polygon for Example \ref{['Ex. FourGens']}

Theorems & Definitions (28)

  • Theorem 1.1: The prime ideal factorization of $2{\mathcal{O}}_{\mathbb{Q}(\space\sqrt[n]{a})}$
  • Theorem 1.2: The prime ideal factorization of $p{\mathcal{O}}_{\mathbb{Q}(\space\sqrt[n]{a})}$
  • Corollary 1.3: Classification of 2 as a common $N$-index divisor
  • Corollary 1.4: Classification of $p$ as a common $N$-index divisor
  • Theorem 2.1
  • Theorem 2.2: The Vahlen--Capelli Theorem
  • Theorem 3.1: Theorem 7 of VelezPrimePower
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 18 more