[Paper Review] On the index divisors and monogenity of certain nonic number fields

Omar Kchit|arXiv (Cornell University)|Jul 6, 2023

Analytic Number Theory Research4 citations

TL;DR

This paper provides a complete characterization of prime divisors of the index $ i(K) $ for nonic number fields $ K $ generated by a root of a monic irreducible trinomial $ x^9 + ax + b \in \mathbb{Z}[x] $. Using Newton polygon methods and $ p $-adic valuations, it determines for each prime $ p $ when $ p \mid i(K) $, computes $ \nu_p(i(K)) $, and shows that $ i(K) \neq 1 $ implies $ K $ is not monogenic. The key contribution is a complete solution to Problem 22 of Narkiewicz for this class of fields.

ABSTRACT

In this paper, for any nonic number field $K$ generated by a root $α$ of a monic irreducible trinomial $F(x)=x^9+ax+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz (\cite{Nar}) for this family of number fields. In particular if $i(K) eq 1$, then $K$ is not mongenic. We illustrate our results by some computational examples.

Motivation & Objective

• To determine for which rational primes $ p $ the index $ i(K) $ of a nonic number field $ K $ is divisible by $ p $, when $ K $ is defined by a monic irreducible trinomial $ x^9 + ax + b \in \mathbb{Z}[x] $.
• To compute the $ p $-adic valuation $ \nu_p(i(K)) $ for each prime $ p $, especially for $ p = 2, 3 $, and $ p \geq 5 $.
• To provide a complete characterization of the prime power decomposition of $ i(K) $, thereby solving a special case of Problem 22 from Narkiewicz's list.
• To determine when such fields are monogenic (i.e., $ i(K) = 1 $) and when they are not, based on the index being nontrivial.
• To illustrate the theory with explicit computational examples where $ i(K) $ is computed as 1, 2, 3, 6, 8, or 24.

Proposed method

• Use of Newton polygon theory over $ \mathbb{Q}_p $ to analyze the factorization of $ F(x) = x^9 + ax + b $ modulo $ p $, particularly for $ p = 2 $ and $ p = 3 $.
• Application of the theorem of Dedekind and the theorem of Ore to relate the factorization of $ p\mathbb{Z}_K $ to the index $ i(K) $.
• Computation of the discriminant $ \Delta $ of $ F(x) $, and its $ p $-adic valuation $ \nu_p(\Delta) $, to determine the behavior of $ p $ in $ \mathbb{Z}_K $.
• Use of the condition $ \nu_p(2^{24}a^9 + 3^{18}b^8) = 1 $ when $ \nu_p(ab) = 0 $ to test integrality of $ \mathbb{Z}[\alpha] $ and monogenity.
• Evaluation of $ \nu_p(i(K)) $ via the structure of the Newton polygon and the number of residue degree 1 prime ideals above $ p $.
• Use of modular conditions on $ (a,b) \mod 4 $, $ \mod 9 $, $ \mod 81 $, and $ \mod 243 $ to determine $ \nu_2(i(K)) $ and $ \nu_3(i(K)) $.

Experimental results

Research questions

• RQ1 For a nonic number field $ K $ defined by $ x^9 + ax + b \in \mathbb{Z}[x] $, which rational primes $ p $ divide the index $ i(K) $?
• RQ2 What is the exact value of $ \nu_p(i(K)) $ for $ p = 2 $, $ p = 3 $, and $ p \geq 5 $, in terms of the coefficients $ a $ and $ b $?
• RQ3 Under what conditions is $ \mathbb{Z}[\alpha] $ integrally closed, and when is $ K $ monogenic?
• RQ4 How can the prime power decomposition of $ i(K) $ be fully determined for this class of fields?
• RQ5 Can the index $ i(K) $ be explicitly computed for specific $ a $ and $ b $, and when is it greater than 1?

Key findings

• For $ a = 51 $, $ b = 122 $, all index conditions in Theorem 2.1 are satisfied, so $ i(K) = 1 $, and $ K $ is monogenic.
• For $ a = 35 $, $ b = 20 $, the condition $ (a,b) \equiv (3,4) \mod 8 $ implies $ \nu_2(i(K)) \geq 1 $, so $ i(K) $ is even and $ K $ is not monogenic.
• For $ a = 1392 $, $ b = 768 $, $ \nu_2(i(K)) = 1 $ and $ \nu_3(i(K)) = 0 $, so $ i(K) = 2 $, and $ K $ is not monogenic.
• For $ a = 126 $, $ b = 40130 $, $ \nu_3(i(K)) = 1 $ and $ \nu_2(i(K)) = 0 $, so $ i(K) = 3 $, and $ K $ is not monogenic.
• For $ a = 15381 $, $ b = 6634 $, $ \nu_2(i(K)) = 1 $ and $ \nu_3(i(K)) = 1 $, so $ i(K) = 6 $, and $ K $ is not monogenic.
• For $ a = 183 $, $ b = 296 $, $ \nu_2(i(K)) = 3 $ and $ \nu_3(i(K)) = 0 $, so $ i(K) = 8 $, and $ K $ is not monogenic.
• For $ a = 7335 $, $ b = 24184 $, $ \nu_2(i(K)) = 3 $ and $ \nu_3(i(K)) = 1 $, so $ i(K) = 24 $, and $ K $ is not monogenic.