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On ideal class groups of totally degenerate number rings

Ruben Hambardzumyan, Mihran Papikian

TL;DR

This work analyzes the ideal class theory of totally degenerate number rings R = Z[x]/(χ(x)) where χ has distinct integer roots. It establishes a Latimer–MacDuffee–style bijection between the ideal class monoid and conjugacy classes of integer matrices, derives an explicit asymptotic for the size of the ideal class monoid, and computes the exact class group size via a class-number formula for orders in product fields. The authors provide Brauer–Siegel–type limits and fully describe the structures for n=2,3, including concrete quadratic and cubic case descriptions. These results illuminate the arithmetic of orders in products of number fields and connect combinatorial matrix data to ideal-theoretic invariants.

Abstract

Let $χ(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring $\mathbb{Z}[x]/(χ(x))$. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of $χ(x)$ tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of $χ(x)$ is $2$ or $3$.

On ideal class groups of totally degenerate number rings

TL;DR

This work analyzes the ideal class theory of totally degenerate number rings R = Z[x]/(χ(x)) where χ has distinct integer roots. It establishes a Latimer–MacDuffee–style bijection between the ideal class monoid and conjugacy classes of integer matrices, derives an explicit asymptotic for the size of the ideal class monoid, and computes the exact class group size via a class-number formula for orders in product fields. The authors provide Brauer–Siegel–type limits and fully describe the structures for n=2,3, including concrete quadratic and cubic case descriptions. These results illuminate the arithmetic of orders in products of number fields and connect combinatorial matrix data to ideal-theoretic invariants.

Abstract

Let be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring . We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of is or .
Paper Structure (7 sections, 27 theorems, 97 equations)

This paper contains 7 sections, 27 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ideal Class Monoid
  4. Ideal class group
  5. Examples of small degree
  6. Quadratic case
  7. Cubic case

Key Result

Theorem 1.1

If $a_n-a_1\geq 4$, then If $a_n-a_1<4$, then $\lvert\mathop{\mathrm{Cl}}\nolimits(R)\rvert = 1$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 44 more