Index, Prime Ideal Factorization in simplest Quartic Fields and counting their discriminants

2018/01/07 by Mohammed Seddik, Seddik, Mohammed
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1801.02232

openalex publication_date 2018/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the simplest quartic number fields \mathbbK_m defined by the irreducible quartic polynomials x⁴-mx³-6x²+mx+1, where m runs over the positive rational integers such that the odd part of m²+16 is squarefree. In this paper, we study the common index divisor I(\mathbb K_m) and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields \mathbbK_m. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant ≤ x and given index.

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