We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit ε u generally generates the unit group of the quartic order Z[ε u ].
We prove that the relative class number of a nonabelian normal CM-field of degree 2 pq (where p and q are two distinct odd primes) is always greater than four. Not only does this result solve the class number one problem for the nonabelian normal CM-fields of degree 42, but it has also been used elsewhere to solve the class number one problem for the nonabelian normal CM-fields of degree 84.
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