论文标题
关于由$ x^{2^u \ cdot 3^v \ cdot 5^t} -m $定义的某些纯数字段的单基因
On monogenity of certain pure number fields defined by $x^{2^u\cdot 3^v\cdot 5^t}-m$
论文作者
论文摘要
令$ k = \ mathbb {q}（α）$为纯数字段，由一个不可约的多项式$ f（x）= x^{2^u \ cdot 3^v \ cd \ cdot 5^t} -m $，带有$ m \ neq \ neq \ neq \ neq \ pm 1 $ $ $ $ $ $ $ $ $ $ $ uneg和$ u u u u u u，$ u u u u u u，$ u u u u u，$ u u u u u，$ u u u，在本文中，我们研究了$ k $的单基因。我们证明，如果$ m \ not \ equiv 1 \ md4 $，$ m \ not \ equiv \ pm 1 \ md9 $，而$ m \ not \ in \ in \ {\ pm 1，\ pm 7 \ pm 7 \} \ md {25} $，那么$ k $是单一的。但是，如果{$ m \ equiv 1 \ md {4} $}或$ m \ equiv 1 \ md9 $或$ m \ equiv -1 \ md9 $ and $ u = 2k $ for某些奇数$ k $或$ k $或$ u \ u \ ge 2 $ and $ m \ equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv equiv an $ u = 2k $对于某些奇数$ k $或$ u = v = 1 $和$ m \ equiv \ pm 82 \ md {5^4} $，然后$ k $不是单基因。
Let $K = \mathbb{Q} (α) $ be a pure number field generated by a root $α$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v\cdot 5^t}-m$, with $ m \neq \pm 1 $ a square free rational integer, $u$, $v$ and $t$ three positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md4$, $m\not\equiv \pm 1\md9$, and $m\not\in\{\pm 1, \pm 7\}\md{25}$, then $K$ is monogenic. But if {$m\equiv 1\md{4}$} or $m\equiv 1\md9$ or $m\equiv -1\md9$ and $u=2k$ for some odd integer $k$ or $u\ge 2$ and $m\equiv 1\md{25}$ or $m\equiv -1\md{25}$ and $u=2k$ for some odd integer $k$ or $u=v=1$ and $m\equiv \pm 82\md{5^4}$, then $K$ is not monogenic.
