Power integral bases for certain pure sextic fields

In this paper, we characterize whether the pure sextic fields [Formula: see text] with square-free integers m ≢ ±1( Mod 9) have power integral bases or do not; if m ≡ 2, 3 (Mod 4), then [Formula: see text] have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic subfields. Based on the works of Kovács and Pethő, several examples on application of monogenic fields to CNS (Canonical Number System) are shown.

INTERIOR COMPONENTS OF A TILE ASSOCIATED TO A QUADRATIC CANONICAL NUMBER SYSTEM — PART II

Let [Formula: see text] be a root of the polynomial p(x) = x2 + 4x + 5. It is well-known that the pair (α, {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each γ ∈ ℤ[α] admits a finite representation of the shape γ = a0 + a1α + ⋯ + aℓαℓ with ai ∈ {0, 1, 2, 3, 4}. The set [Formula: see text] of points with integer part 0 in this number system [Formula: see text] is called the fundamental domain of this canonical number system. It is a plane continuum with nonempty interior which induces a tiling of ℂ. Moreover, it has a disconnected interior [Formula: see text]. In the first paper of this series we described the closures C0, C1, C2 and C3 of the four largest components of [Formula: see text] as attractors of graph-directed self-similar sets. Each of these four sets is a translation of C0. We conjectured that the closures of the other components are images of C0 by similarity transformations. In this article we prove this conjecture. Moreover, we provide a graph from which the suitable similarities can be read off.

A TECHNIQUE IN THE TOPOLOGY OF CONNECTED SELF-SIMILAR TILES

Not much is known about the topological structure of a connected self-similar tile whose interior is disconnected, and even less is understood if the interior consists of infinitely many components. We introduce a technique to show that for a large class of self-similar tiles in ℝ2, the closure of each component of the interior is homeomorphic to a disk. This allows us to prove such a result for the Eisenstein set, the fundamental domain of a well-known quadratic canonical number system, and some other well-known fractal tiles.