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Determination of all quaternion CM-fields with ideal class groups of exponent 2. (English) Zbl 0952.11025
In [Acta Arith. 67, 47-62 (1994; Zbl 0809.11069)], the authors of this paper showed that the class numbers of octic quaternion CM-fields \(N\) (totally complex normal octic fields whose Galois group over the rationals is the quaternion group of order \(8\)) are always even. Here they show that there are exactly two such fields whose class groups have exponent \(2\), namely \(\mathbb Q(\sqrt{-(2+\sqrt{2})(3+\sqrt{2})})\) with class number \(2\) and \(\mathbb Q(-\sqrt{5+\sqrt{5})(5+\sqrt{21})(21+2\sqrt{105})})\) with class group of type \((2,2,2)\). The proof consists of two parts:
1) an arithmetic part using class field theory showing that if the class group of \(N\) has vanishing \(4\)-rank, then the maximal real subfield \(K\) of \(N\) has odd class number, and there are at most four primes ramifying in \(K\).
2) an analytic part giving a finite list of fields \(N\) whose maximal real subfields \(K\) satisfy the conditions from 1) and whose \(2\)-class groups are annihilated by \(2\). For each of the fields from this finite list, the class group is computed, and it turns out that only the two fields above have a class group that is an elementary abelian 2-group.

11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions