The Index of a Cyclic Quartic Field

Abstract

Let K be a cyclic quartic field. Let i(K) denote the index of K. It is known that i(K)∈{1, 2, 3, 4, 6, 12}. In Part 1 of this paper we show that i(K) assumes all of these values and we give necessary and sufficient conditions for each to occur. In Part 2 an asymptotic formula is given for the number of cyclic quartic fields with discriminant ≤x and i(K)=i for each i∈{1, 2, 3, 4, 6, 12}.