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

Abstract

We consider the simplest quartic number fields K m defined by the irreducible quartic polynomialsx 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(K m ) and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields K m . On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant ≤ x and given index.