On monogenity of certain pure number fields defined by $x^{2^u\cdot 3^v\cdot 5^t}-m$

利用椭圆曲线确定纯三次数域的单源性

Determining monogenity of pure cubic number fields using elliptic curves

该研究利用椭圆曲线的 3-同源 Selmer 群刻画纯三次数域单源性，提出排除判据并给出单源域数量界，且在 GRH 下判定多个具体域。

关于带非平方自由参数的纯八次数域的整基与单本性

On integral bases and monogenity of pure octic number fields with non-square free parameters

本文研究了非平方自由参数下纯八次数域的单本性，通过计算其整数环的整基，刻画了整基等于 $\mathbf Z[α]$ 的条件，并给出了数域非单本的充分条件及特殊情形分析。

On index divisors and monogenity of certain number fields defined by trinomials

Let $K$ be a number field generated by a root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we study the problem of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree $n=2^r\cdot3^k$ with $r$ and $k$ two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this pape…

全复纯八次域的单演性

On the monogenity of totally complex pure octic fields

本文研究了全复纯八次域的幂积分基生成元问题，提出了一种高效计算方法，并在 $0>m>-5000$ 范围内的 2024 个域中验证了是否存在与 $α$ 不等价的生成元。

关于由四项式定义的四次数域的指数与单基因性

On indices and monogenity of quartic number fields defined by quadrinomials

本文研究由不可约四项式生成的四次数域的指数，给出其被2或3整除的充分条件并确定指数中2、3的精确幂次，同时构造出新的无限族单基因四次数域。

On power integral bases of certain pure number fields defined by $x^{2^u\cdot3^v}-m$

Let $K = \mathbb{Q} (α) $ be a pure number field generated by a complex root $α$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v}-m$, with $m \neq \pm 1$ a square free rational integer, $u$, and $v$ two positive integers. In this paper, we study the monogenity of $K$. The case $u=0$ or $v=0$ has been previously studied. We prove that if $m\not\equiv 1$ (mod4) and $m\not\equiv \pm 1$ (mod9), then $K$ is monogenic. But if $m\equiv 1$ (mod4) or $m\equiv 1$ (mod9), then $K$ is not monogenic. Some illustrating examples are given too.

关于由 $x^n +ax^m+b$ 型三项式定义的非单生成数域

On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$

本文研究了由 $x^n+ax^m+b$ 型三项式定义的数域的非单生性问题，给出了判定非单生成的显式条件，并证明了存在无限族的非单生成数域。

多项式的合成指数及其应用

The Index of Composition of polynomials and its applications

本文研究首一不可约多项式的合成指数，证明指数整除性并给出精确幂次，进而应用于判定多项式迭代的单生成性及数域塔中的单生成数域个数。

纯数域整数基的最小周期性

The minimal periodicity for integral bases of pure number fields

本文研究了纯数域中整数基的最小周期性，通过局部到全局的原则，证明了全局形状的周期模为 $M(n)=n\cdot\mathrm{rad}(n)$，并给出了应用。

Noether's problem for some subgroups of $S_{14}$: the modular case

Let $G$ be a subgroup of $S_{n}$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $σ\cdot x_{i}:=x_{σ\cdot i}$ for any $σ\in G$ and $1\leq i\leq n$. In this article, we will show that if $G$ is a solvable transitive subgroup of $S_{14}$ and $\text{char}(k)=7$, then the fixed subfield $k(x_{1},\cdots,x_{14})^{G}$ is rational (i.e., purely transcendental) over $k$. In proving the above theorem, we rely on the Kuniyoshi-Gaschütz Theorem or some ideas in its proof.

通过欧几里得算法处理某些丢番图方程及其在纯指数方程中的应用

Handling some Diophantine equation via Euclidean algorithm and its application to purely exponential equations

本文通过经典和新方法研究三元指数丢番图方程，验证了R.Scar和R.Styer猜想在特定条件下成立，并应用于多项式-指数丢番图方程。

Noether's Problem for Some Semidirect Products

Let $k$ be a field, $G$ be a finite group, $k(x(g):g\in G)$ be the rational function field with the variables $x(g)$ where $g\in G$. The group $G$ acts on $k(x(g):g\in G)$ by $k$-automorphisms where $h\cdot x(g)=x(hg)$ for all $h,g\in G$. Let $k(G)$ be the fixed field defined by $k(G):=k(x(g):g\in G)^G=\{f\in k(x(g):g\in G): h\cdot f=f$ for all $h\in G\}$. Noether's problem asks whether the fixed field $k(G)$ is rational (= purely transcendental) over $k$. Let $m$ and $n$ be positive integers and assume that there is an integer $t$ such that $t\in (\bm{Z}/m\bm{Z})^\times$ is of order $n$. Defi…

Noether's problem for some 2-groups

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\in G)^G$ is rational (i.e. purely transcendental) over $k$. We will prove that, if $G$ is a group of order $2^n$ ($n\ge 4$) and of exponent $2^e$ such that (i) $e\ge n-2$ and (ii) $ζ_{2^{e-1}} \in k$, then $k(G)$ is $k$-rational.13A50,14E08,14M20,12F12