This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite set of arbitrary size may ramify. We then continue by examining a conjecture of Harbater that the minimal number of generators of the Galois group of a tame, Galois extension of the rational numbers is bounded above by the sum of a constant and the logarithm of the product of the ramified primes. We prove the validity of Harbater's conjecture in a number of cases, including the situation where we restrict our attention to finite groups containing a nilpotent subgroup of index $1,2,$ or $3$. We also derive some consequences that are implied by the truth of this conjecture.

中文翻译：

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逆伽罗瓦问题的分枝

本文重点介绍逆伽罗瓦问题的改进。我们探索什么有限群作为有理数扩展的伽罗瓦群出现，其中只有一组预定的素数可以分枝。在展示了关于只有一个有限质数分枝的扩展的新结果之后，我们继续研究更复杂的情况，其中来自任意大小的有限集合的多个质数可能会分枝。然后，我们继续检查 Harbater 的一个猜想，即有理数的伽罗瓦扩展的最小数量的生成器的最小数量由常数和分枝素数的乘积的对数之和限定。我们在许多情况下证明了 Harbater 猜想的有效性，包括我们将注意力限制在包含指数为 $1,2,$ 或 $3$ 的幂零子群的有限群的情况。我们还推导出了这个猜想的真实性所暗示的一些后果。