Soun-Hi Kwon

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 possible fields of degree 64 or 96. Consequently the class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.

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

We prove that there is precisely one normal CM-field of degree 48 with class number one which has a normal CM=subfield of degree 16: the narrow Hilbert class field of Q(√5, √101, θ) with θ3 - θ2 -5θ - 1 = 0.

THE CLASS GROUPS OF THE IMAGINARY ABELIAN NUMBER FIELDS WITH GALOIS GROUP (Z/2Z) n

Assuming the Generalised Riemann Hypothesis we determine all imaginary Abelian number fields N whose Galois group G ( N /ℚ) is isomorphic to (ℤ/2ℤ) n for some integers n ≥ 1 and the square of every ideal of N is principal.

The imaginary abelian number fields of 2-power degrees with ideal class groups of exponent ≤2

In this paper, assuming the Generalized Riemann Hypothesis, we determine all imaginary abelian number fields N of 2-power degrees with ideal class groups of exponents ≤ 2 for which the 2-ranks of the Galois group of N over Q are equal to 2.

NONABELIAN NORMAL CM-FIELDS OF DEGREE 2 pq

We prove that the relative class number of a nonabelian normal CM-field of degree 2pq (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. 2000 Mathematics subject classification: primary 11R29; secondary 11R20.