论文标题
关于由$ x^{2^r}+ax^m+b $定义的某些数字字段的共同索引分隔和单个字段的单个字段
On common index divisors and monogenity of certain number fields defined by trinomials of type $x^{2^r}+ax^m+b$
论文作者
论文摘要
令$ k = \ q（Þ）$为$ $ $ $ f（x）= x^{2^r}+ax^m+b \ in \ z [x] $的数字。在本文中，基于$ p $ adic牛顿多边形技术，用于数字字段中的素数分解和ore \ cite \ cite {narprime，o}的经典索引定理，我们研究$ k $的单一性。更确切地说，我们证明，如果$ a $和$ 1+b $都可以分别为$ 32 $，那么$ k $就不可能是单一的。对于$ m = 1 $，我们提供$ a $，$ b $和$ r $的明确条件，$ k $不是单基因。我们还构建了一个不可减少的三项式家族，这些家族不是单基因的，但是它们的根部产生了单基数字段。为了说明我们的结果，我们提供了一些计算示例。
Let $K = \Q(þ)$ be a number with $þ$ a root of an irreducible trinomial of type $ F(x)= x^{2^r}+ax^m+b \in \Z[x]$. In this paper, based on the $p$-adic Newton polygon techniques applied on decomposition of primes in number fields and the classical index theorem of Ore \cite{Narprime, O}, we study the monogenity of $K$. More precisely, we prove that if $a$ and $1+b$ are both divisible by $32$, then $K$ cannot be monogenic. For $m=1$, we provide explicit conditions on $a$, $b$ and $r$ for which $K$ is not monogenic. We also construct a family of irreducible trinomials which are not monogenic, but their roots generate monogenic number fields. To illustrate our results, we give some computational examples.
