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arXiv · 1801.02232

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

Mohammed Seddik
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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.

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Mohammed Seddik. 2018-01-07. Index, Prime Ideal Factorization in simplest Quartic Fields and counting their discriminants. https://arxiv.org/abs/1801.02232

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