Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible $\ell$-adic representation of $G_K$ of dimension $d$ with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic of $K$ is $>3$. The result stated in the title is a special case.

中文翻译：

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Qp 上的每个 7 维阿贝尔变体都有一个可约的 ℓ-adic Galois 表示

让 $钾$ 是具有有限残差域的完全离散值域，并且 $G_{钾}$其绝对伽罗瓦群。本笔记的主题是研究正整数集$d$ 存在一个绝对不可约的 $\ell$-adic 表示 $G_{钾}$ 维度的 $d$在惯性上有理性的痕迹。我们的主要结果是，当残差特征为$钾$ 是 $>3$. 标题中所述的结果是一个特例。