论文标题
代数数字的分母中的素数
Primes in denominators of algebraic numbers
论文作者
论文摘要
将代数数字表示为$ \ overline {\ mathbb {q}} $和代数整数的集合为$ \ overline {\ mathbb {z}} $。对于$γ\ in \ overline {\ mathbb {q}} $，请考虑其在$ \ mathbb {z} [x] $，$f_γ（x）= a_nx^n+\ dots+dots+a_0 $中的$ \ mathbb {z} [x] $中的多项式。表示$ e（γ）= \ gcd（a_ {n}，a_ {n-1}，\ dots，a_1）$。 Drungilas，Dubickas和Jankauskas在最近的一篇论文中表明，$ \ Mathbb {Z} [γ] \ Cap \ Mathbb {q} = \ {α\ in \ in \ Mathbb {q} \ nim Mid \ MID \ MID \ \ MID \ {p \ Mid v_p（α） p | e（γ）\} \} $。给定一个数字字段$ k $和$γ\ in \ in \ overline {\ mathbb {q}} $，我们表明存在一个子集$ x（k，γ）\ subseteq \ subseteq \ text {spec}（\ m马理{o} _k）$ k \ mid \ {\ mathfrak {p} \ mid v _ {\ mathfrak {p}}}（α）<0 \} \ subseteq x（k，γ）\} $。我们证明$ \ MATHCAL {O} _K [γ] \ cap K $是主要理想域，并且仅当$ x（k，γ）$中的总数生成$ \ Mathcal {o} o} _k $的类组。我们证明给定的$γ\ in \ int \ overline {\ mathbb {q}} $，我们可以找到一个有限的集合$ s \ subseteq \ subseteq \ overline {\ mathbb {z}} $，因此对于每个数字$ k $，我们都有$ x（k，γ）= \ {\ mathfrak {p} \ in \ text {spec}（\ mathcal {o} _k）\ mid \ mathfrak {p} \ cap s \ cap s \ neq \ neq \ emptyset \ \} $。我们研究了此$ s $如何与环$ \ overline {\ mathbb {z}} [γ] $和理想的$ \ mathfrak {d}_γ= \ {a \ in \ in \ In \ intline { $ \ OVILLINE {\ MATHBB {Z}} $。我们还表明，$γ_1，γ_2\ in \ edline {\ mathbb {q}} $满足$ \ mathfrak {d} _ {γ_1} = \ mathfrak {d} _ {d} _ {d} _ {γ_2} $ if，并且只有$ x（k，k，k，k，g，γ_1）$ k，$ k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，
Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $γ\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$, $F_γ(x)=a_nx^n+\dots+a_0$. Denote $e(γ)=\gcd(a_{n},a_{n-1},\dots,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $\mathbb{Z}[γ]\cap \mathbb{Q}=\{α\in\mathbb{Q}\mid \{p\mid v_p(α)<0\}\subseteq \{p\mid p|e(γ)\}\}$. Given a number field $K$ and $γ\in\overline{\mathbb{Q}}$, we show that there is a subset $X(K,γ)\subseteq \text{Spec}(\mathcal{O}_K)$, for which $\mathcal{O}_K[γ]\cap K=\{α\in K\mid \{\mathfrak{p}\mid v_{\mathfrak{p}}(α)<0\}\subseteq X(K,γ)\}$. We prove that $\mathcal{O}_K[γ]\cap K$ is a principal ideal domain if and only if the primes in $X(K,γ)$ generate the class group of $\mathcal{O}_K$. We show that given $γ\in \overline{\mathbb{Q}}$, we can find a finite set $S\subseteq \overline{\mathbb{Z}}$, such that for every number field $K$, we have $X(K,γ)=\{\mathfrak{p}\in\text{Spec}(\mathcal{O}_K)\mid \mathfrak{p}\cap S\neq \emptyset\}$. We study how this set $S$ relates to the ring $\overline{\mathbb{Z}}[γ]$ and the ideal $\mathfrak{D}_γ=\{a\in\overline{\mathbb{Z}}\mid aγ\in\overline{\mathbb{Z}}\}$ of $\overline{\mathbb{Z}}$. We also show that $γ_1,γ_2\in \overline{\mathbb{Q}}$ satisfy $\mathfrak{D}_{γ_1}=\mathfrak{D}_{γ_2}$ if and only if $X(K,γ_1)=X(K,γ_2)$ for all number fields $K$.
