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Let be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form whose ideal class group has an element of order . This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups.
We characterize 2-birational CM-extensions of totally real number fields in terms of tame ramification. This result completes in this case a previous work on pro-l-extensions over 2-rational number fields.
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