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On a weak form of Ennola's conjecture about certain cubic number fields

Jinwoo Choi, Dohyeong Kim

Abstract

We establish a weak form of Ennola's conjecture. We achieve this by showing that two main assumptions Louboutin made in his previous work hold true. These assumptions are about Laurent polynomials over the rationals, and we prove them by using Newton identities.

On a weak form of Ennola's conjecture about certain cubic number fields

Abstract

We establish a weak form of Ennola's conjecture. We achieve this by showing that two main assumptions Louboutin made in his previous work hold true. These assumptions are about Laurent polynomials over the rationals, and we prove them by using Newton identities.

Paper Structure

This paper contains 13 sections, 5 theorems, 72 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Louboutin's conjectures
  3. Proof of Conjecture 1
  4. Proof of Conjecture 2
  5. Case 1: $a \geq 1$ odd and $b \geq 1$ odd
  6. $\max(a,b)=a$
  7. $\max(a,b)=b$
  8. Case 2: $a \geq 1$ odd and $b \geq 1$ even
  9. $\max(a,b)=a$
  10. $\max(a,b)=b$
  11. Case 3: $a \geq 2$ even and $b \geq 1$ odd
  12. $\max(a,b)=a$
  13. $\max(a,b)=b$

Key Result

Theorem 1

For any given prime $p \geq 3$, there are only finitely many $l \geq 3$ for which $p$ divides the unit index $j_l$. Hence, for any given integer $N \geq 2$ we have $\gcd (j_l, N!) = 1$ for $l \geq l_N$ effectively large enough.

Theorems & Definitions (13)

  • Theorem 1: weak form of Ennola's conjecture, Theorem 2 of Lou21
  • Definition 1
  • Example 1
  • Conjecture 1: Conjecture 14 of Lou21
  • Proposition 1: Proposition 19 of Lou21
  • Conjecture 2: Conjecture 20 of Lou21
  • Theorem 2: Newton identities, Exercise 14 in $\S 7.1$ of CLO
  • Example 2
  • Proposition 2
  • proof
  • ...and 3 more