论文标题
在由$ x^n +ax^m +b $类型定义的非单性数字字段上
On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$
论文作者
论文摘要
令$ k = \ q（θ）$为一个数字字段，该数字是由一个不可约束的三位一体$ f（x）= x^n+ax^{m}+b \ in \ z [x] $生成的。在本文中，我们处理了$ k $的非单据基因性问题。更确切地说，我们为$ a $，$ b $，$ n $和$ m $提供一些明确的条件，$ k $不是单基因。作为应用程序，我们表明，由$ n = 2^r \ cdot3^k $定义的无单性数字字段的无限家族，带有$ r $，$ k $是正整数。我们还提供了两个无单性数字领域的无限家族，该领域由$ 6 $ $ 6 $定义。最后，我们通过举例说明了结果。
Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we deal with the problem of the non-monogenity of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As application, 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 $k$ are positive integers. We also give two infinite families of non-monogenic number fields defined by trinomials of degree $6$. Finally, we illustrate our results by giving some examples.
