Cyclic polytope of the simplest cubic fields

HT90 and "simplest" number fields

A standard formula (1) leads to a proof of HT90, but requires proving the existence of $θ$ such that $α\ne 0$, so that $β=α/σ(α)$. We instead impose the condition (M), that taking $θ=1$ makes $α=0$. Taking $n=3$, we recover Shanks's simplest cubic fields. The "simplest" number fields of degrees 3 to 6, Washington's cyclic quartic fields, and a certain family of totally real cyclic extensions of $\Q(\cos(π/4m))$ all have defining polynomials whose zeroes satisfy \eqref{e:M}. Further investigation of (M) for $n=4$ leads to an elementary algebraic construction of a 2-parameter family of octic pol…

The characterization of cyclic cubic fields with power integral bases

We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. We show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ which is the splitting field of a Shanks cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbb Z$. Moreover we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.

On the K-theory of truncated polynomial algebras, revisited

We revisit the computation, due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras for perfect fields of positive characteristic. The resulting K-groups are expressed in terms of big Witt vectors of the field. The original proof relied on an understanding of cyclic polytopes in order to determine the genuine equivariant homotopy type of the cyclic bar construction for a suitable monoid. Using the Nikolaus-Scholze framework for topological cyclic homology we achieve the same result using only the homology of said cyclic bar construction, as well as the action of Connes' …

Integral bases and monogenity of the simplest sextic fields

Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $α$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \] The totally real cyclic fields $K=Q(α)$ are called simplest sextic fields and are well known in the literature. Using a completely new approach we explicitly give an integral basis of $K$ in a parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$ in which cases we give all generators of power integral bases.

最简三次域与连续双二次域中的大Pólya群

Large Pólya groups in simplest cubic fields and consecutive bi-qaudratic fields

本文将数域的连续性概念推广到循环三次域和双二次域的参数族，证明存在无穷多个具有大Pólya群的连续域，推广了全实双二次域的已有结果。

高亏格曲线上的循环三次点

Cyclic Cubic Points on Higher Genus Curves

本文研究高亏格曲线上伽罗瓦群为 $C_3$ 的三次点，证明其源于结构良好的态射，并给出可计算检验，同时推广至低亏格情形。