Non-Galois cubic number fields with exceptional units. Part II

Exceptional units in a family of quartic number fields

We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a oneparameter family of polynomials with two real roots.

Class-number Problems for Cubic Number Fields

Chebyshev Covers and Exceptional Number Fields

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, Tk(x) and Uk(x). In our normalization, the indices are allowed to be halfintegers as well as integers, and the “polynomials” actually live in Z[x, √ 2− x, √ 2 + x]. We will show that the rational functions Tm/2(x) Tn/2(x) and Um/2(x) Un/2(x) are very remarkable from the point of view of G...

Purely Cubic Complex Function Fields With Small Units

We investigate several infinite families of purely cubic complex congruence function fields with small fundamental units. Specifically, we compute the fundamental units of fields K of unit rank 1 and characteristic not equal to 3 where the generator of K over Fq(t) is a cube root of D = (M3 − F )/E3 with E3 dividing M3 − F and F dividing M2. We also characterize all purely cubic complex functio...

Procyclic Galois Extensions of Algebraic Number Fields

6 1 Iwasawa’s theory of Zp-extensions 9 1.