## Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places?

See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a set $S$ of places where $S$ is not finite (for every $\dim{V} \geq 1$)?

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For $\mathrm{dim}(V) = 2$, see: jstor.org/stable/40067979. For $\mathrm{dim}(V) \ge 2$, take symmetric powers. – Lavender Honey Aug 18 at 20:38
@Timo: I've added a tag and would also urge you to incorporate your own "answer" into an edited question to avoid confusion and focus the question better. – Jim Humphreys Aug 19 at 23:00

It seems that for $\dim{V} = 1$, there is no such representation http://www.math.leidenuniv.nl/scripties/KretMaster.pdf p. 10.