关于由三项式定义的某些数域的单整性
On monogenity of certain number fields defined by trinomials
摘要：
设$K=\Q(θ)$是由一元不可约三项式$F(x) = x^n+ax+b \in \Z[x]$的复根$þ$生成的数域。关于由三项式定义的数域的单整性有大量的文献，Gaál研究了由三项式定义的六次数域的多重单整性。Jhorar和Khanduja研究了$\Z[þ]$的积分闭性。但如果$ \Z[þ]$不是整闭的，则Jhorar和Khanduja的结果不能回答$K$的单纯性.本文利用牛顿多边形技术，研究了$K$的单正则性问题.更准确地说，当$\Z_K \neq \Z[þ]$，我们给出了充分条件$n$，$a$和$b$的$K$是不是单基因的。对于$n\in \{5, 6, 3^r, 2^k\cdot 3^r, 2^s\cdot 3^k+1\}$，我们明确给出了这类数域的非单演无穷族。最后，我们通过一些计算实例来说明我们的结果。
英文摘要：
Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of $\Z[þ]$. But if $ \Z[þ]$ is not integrally closed, then Jhorar and Khanduja's results cannot answer on the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of $K$. More precisely, when $\Z_K \neq \Z[þ]$, we give sufficient conditions on $n$, $a$ and $b$ for $K$ to be not monogenic. For $n\in \{5, 6, 3^r, 2^k\cdot 3^r, 2^s\cdot 3^k+1\}$, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.
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