We give a streamlined and effective proof of Ozaki’s theorem that any finite $p$-group $\Gamma$ is the Galois group of the $p$-Hilbert class field tower of some number field $F<!--FUNCTION\; APPLICATION-->$. 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 ${k<!--FUNCTION\; APPLICATION-->}_0$ with class number prime to $p$. We construct $F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$ by a sequence of $\mathbb{Z}∕p$-extensions ramified only at finite tame primes and also give explicit bounds on $[F<!--FUNCTION\; APPLICATION-->:{k<!--FUNCTION\; APPLICATION-->}_0]$ and the number of ramified primes of $F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$ in terms of $\#\Gamma$.

中文翻译：

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论将规定 p 群实现为 p 级塔群的尾崎定理

我们给出了尾崎定理的简化且有效的证明，即任何有限$p$-团体$\gamma$是伽罗瓦群$p$- 某些数场的希尔伯特级场塔$F<!--FUNCTION\; APPLICATION-->$。我们的工作受到尾崎的启发，并适用于更广泛的情况。虽然他的定理是在完全复杂的设置中，但我们在存在数字字段的任何混合签名设置中获得结果${k<!--FUNCTION\; APPLICATION-->}_0$类号为素数$p$。我们构建$F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$通过一系列$\mathbb{Z}∕p$-扩展仅在有限素数上分支，并且还给出了显式界限$[F<!--FUNCTION\; APPLICATION-->:{k<!--FUNCTION\; APPLICATION-->}_0]$以及 的分支素数的个数$F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$按照$\#\gamma$。