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.

Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p ? n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer?s one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in.

We consider the simplest quartic number fields Km 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 square free. In this paper, we study the index I(Km) and determine the explicit prime ideal factorization of rational primes in simplest quartic number fields Km. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant ? x and given index.