论文标题
在Ozaki的定理上，将规定的$ p $ - 群体视为$ P $ - 类塔组组
On Ozaki's theorem realizing prescribed $p$-groups as $p$-class tower groups
论文作者
论文摘要
我们简化了有效的Ozaki定理证明，任何有限的$ p $ -group $γ$都是$ p $ -Hilbert class class Field Tower的Galois集团的某些数字field $ \ rm f $。我们的工作灵感来自奥扎基（Ozaki），并在更广泛的情况下适用。虽然他的定理处于完全复杂的环境中，但我们以任何混合签名设置获得了一个数字字段$ {\ rm k} _0 $，class number prime到$ p $。我们构造$ {\ Mathbb z}/p $ - extensions仅在有限的驯高素数上对$ {\ Mathbb z}/p $ - extensions构造$ {\ rm f}/{\ rm k} _0 $ $ f}/{\ rm k} _0 $以$ \＃γ$表示。
We give a streamlined and effective proof of Ozaki's theorem that any finite $p$-group $Γ$ is the Galois group of the $p$-Hilbert class field tower of some number field $\rm F$. Our work is inspired by Ozaki's and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field ${\rm k}_0$ with class number prime to $p$. We construct ${\rm F}/{\rm k}_0$ by a sequence of ${\mathbb Z}/p$-extensions ramified only at finite tame primes and also give explicit bounds on $[{\rm F}:{\rm k}_0]$ and the number of ramified primes of ${\rm F}/{\rm k}_0$ in terms of $\# Γ$.
