The class number one problem for the normal CM-fields of degree 32

Abstract

Abstract. We prove that there are exactly six normal CM-fields of degree 32 with class number one. Five of them are composita of two normal CM-fields of degree 16 with the same maximal totally real octic field.

Normal CM-fields with class number one

“…A common strategy is to use the subfield structure of L or analytic techniques pioneered by Louboutin in [25], to obtain lower bounds on the relative class number of L. This is then used to build a small list of candidate fields L, for which the relative class number is computed using knowledge about the Hasse unit index of L and an algorithm of Louboutin (see [27,28]). For example, in [36] this strategy is used to determine in an ad-hoc fashion for each of the 44 non-abelian groups of order 32 the normal CM-fields with (relative) class number one.…”

“…For d ∈ {20, 40}, the work of [22] and Park [37] solves the class number one problem. For d = 36 the problem was solved by Chang-Kwon [7] and for d = 48 by Chang-Kwon [8] and Park-Kwon [36]. The case d = 32 was settled by Park-Yang-Kwon [36].…”

“…For d = 36 the problem was solved by Chang-Kwon [7] and for d = 48 by Chang-Kwon [8] and Park-Kwon [36]. The case d = 32 was settled by Park-Yang-Kwon [36]. Finally, for d ∈ {1, .…”

“…. , 96} \ {64, 96}, the remaining cases were settled by Park-Kwon [36]. Thus assuming GRH, except for the possible degrees 64 and 96, all normal CM-fields with class number one have been determined.…”

Class number one problem for normal CM-fields

“…All normal CM-fields of degrees less than 48 and class number one are known. For full details see [8,16,17,19,20,[29][30][31][32]38,42]. For those of degree 48 the problem is partially solved: there is precisely one normal CM-field of degree 48 with class number one which has a normal CM-subfield of degree 16 [9].…”