Ryotaro Okazaki

The class number one problem for some non-abelian normal CM-fields

Let N be a non-abelian normal CM-eld of degree 4p; p any odd prime. Note that the Galois group of N is either the dicyclic group of order 4p; or the dihedral group of order 4p: We prove that the (relative) class number of a dicyclic CM-eld of degree 4p is always greater then one. Then, we determine all the dihedral CM-elds of degree 12 with class number one: there are exactly nine such CM-elds.

Similar dissection of sets

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner’s questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let

Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one

{D\subset \mathbb{R}^d}

On the number of solutions of simultaneous Pell equations II

be a given set and let f1, . . . , fk be injective continuous mappings. Does there exist a set X such that

Quartic Thue Equations

{D = X \cup f_1(X) \cup \cdots \cup f_k(X)}

Exponents of the ideal class groups of CM number fields

is satisfied with a non-overlapping union? We will prove that such a set X exists for certain choices of D and {f1, . . . , fk}. The solutions X will often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner’s setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio 1 : 1 : a for