THE 2-RANK OF THE REAL PURE QUARTIC NUMBER FIELD K=ℚ(pd24)

Abstract

. In this paper, we consider the real pure quartic number field K = Q ( 4 (cid:112) pd 2 ) , where p is a prime number and d is a square-free positive integer such that d is prime to p . We compute r 2 ( K ) the 2 -rank of the class group of K and as an application we exhibit all possible forms of d for which the 2 -class group of K is trivial (equivalently: the class number of K is odd), cyclic or isomorphic to Z / 2 n 1 Z × Z / 2 n 2 Z , where n i ∈ N ∗ .