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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.

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