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

TL;DR

This paper investigates the structure and prime ideal decomposition in simplest quartic fields defined by specific polynomials, providing explicit factorizations and an asymptotic count of such fields with bounded discriminant.

Contribution

It explicitly determines prime ideal decompositions and the common index divisor in simplest quartic fields, and establishes an asymptotic formula for counting these fields by discriminant.

Findings

Explicit prime ideal decomposition in simplest quartic fields.

Determination of the common index divisor for these fields.

Asymptotic formula for the number of fields with bounded discriminant.

Abstract

We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ 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 $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.