Index, the prime ideal factorization in simplest quartic fields and counting their discriminants

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 square free.
In this paper, we study the index $I(\mathbb K_m)$ and determine the explicit prime ideal factorization of rational primes in 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.