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The non-normal quartic CM-fields and the dihedral octic CM-fields with ideal class groups of exponent $$\leq 2$$. (English) Zbl 1108.11085
Summary: After having proved that there are only finitely many of them, we determine all the non-normal quartic CM-fields whose ideal class groups have exponent $$\leq 2$$. There are 678 non-isomorphic such quartic CM-fields and 37 out of them have class number 1, 205 out of them have class number 2, 284 out of them have class number 4, 140 out of them have class number 8 and 12 out of them have class number 16. We then deduce that there are 116 dihedral octic CM-fields whose ideal class groups have exponent $$\leq 2$$ and 17 out of them have class number 1, 7 out of them have class number 2, 50 out of them have class number 4, 31 out of them have class number 8, 3 out of them have class number 16, 3 out of them have class number 32, and 5 out of them have class number 64.

##### MSC:
 11R29 Class numbers, class groups, discriminants 11Y40 Algebraic number theory computations 11R37 Class field theory
##### Keywords:
CM-field; class group; quartic field; octic field
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##### References:
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