For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$. For these groups $G$, this gives the existence of number fields of degree $ge(G)$ with an unramified $G$-extension. The existence of such extensions over $\mathbb{Q}$ for all finite groups would imply that, for every finite group $G$, there exists a quadratic number field admitting an unramified $G$-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when $\mathbb{Q}$ is replaced with a function field $k(t)$ where $k$ is an ample field.

中文翻译：

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低度数域上的无分支扩展

对于各种不可解群$G$，我们证明了有理数$\mathbb{Q}$与伽罗瓦群$G$和惯性群除以$ge(G)$的存在性，其中$ge(G)$是 $G$ 的发电机组的最小指数。对于这些组 $G$，这给出了度数为 $ge(G)$ 的数字域的存在，并且具有未分支的 $G$ 扩展。对于所有有限群在 $\mathbb{Q}$ 上的这种扩展的存在意味着，对于每个有限群 $G$，都存在一个二次数域，它允许一个无分枝的 $G$-扩展，正如最近所推测的那样。我们还通过证明当 $\mathbb{Q}$ 被函数域 $k(t)$ 替换其中 $k$ 是一个充足域时它们的存在，为所有有限群存在这种扩展提供了进一步的证据。