The class number one problem for some totally complex quartic number fields

The class number one problem for some non-abelian normal CM-fields of degree 48

We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to A4, the alternating group of degree 4 and order 12. There are two such fields with Galois group A4 × C2 (see Theorem 14) and at most one with Galois group SL2(F3) (see Theorem 18); if the Generalized Riemann Hypothesis is true, the...

On Shintani’s ray class invariant for totally real number fields

We introduce a ray class invariant X(C) for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula X(C) = X1(C) · · ·Xn(C) where each Xi(C) corresponds to a real place (Theorem 3.5). Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices ...

CM-fields with relative class number one

We will show that the normal CM-fields with relative class number one are of degrees ≤ 216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees ≤ 96, and the CM-fields with class number one are of degrees ≤ 104. By many authors all normal CM-fields of degrees ≤ 96 with class number one are known except for the possib...

Class-number Problems for Cubic Number Fields

Units in families of totally complex algebraic number fields

Multidimensional continued fraction algorithms associated with GL n (Z K), where Z k is the ring of integers of an imaginary quadratic field K, are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight. 1. Introduction. Let F be an algebraic number field of degree n. There exist exactly n field embeddin...