Some Unramified Cyclic Cubic Extensions of Pure Cubic Fields

On Euclid's Algorithm in Some Cyclic Cubic Fields

We now let # be the set of points P o such that M = M(P0); if sup M(Pt) < M Pit* (which happens in all cases known so far) we call this the second minimum, M2. R(0) is Euclidean if, and only if, M{P) < 1 for all rational triads [x,y,z]; and if 4> and \jj can be expressed as quadratic polynomials in 0, then the field is said to be cyclic. Heilbronn [1] has shown that Euclid's Algorithm holds in ...

The shapes of pure cubic fields

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are rectangular and that they are equidistributed, in a regularized sense, when ordered by discriminant, in the one-dimensional space of all rectangular lattices. W...

Tame kernels of cubic cyclic fields

There are many results describing the structure of the tame kernels of algebraic number fields and relating them to the class numbers of appropriate fields. In the present paper we give some explicit results on tame kernels of cubic cyclic fields. Table 1 collects the results of computations of the structure of the tame kernel for all cubic fields with only one ramified prime p, 7 ≤ p < 5, 000....

Binomial squares in pure cubic number fields

Let K = Q(ω), with ω3 = m a positive integer, be a pure cubic number field. We show that the elements α ∈ K× whose squares have the form a − ω for rational numbers a form a group isomorphic to the group of rational points on the elliptic curve Em : y2 = x3 − m. This result will allow us to construct unramified quadratic extensions of pure cubic number fields K.

Elliptic units of cyclic unramified extensions of complex quadratic fields