Let $E/K$ be a cubic extension of number fields, $\nu$ be a Grossencharacter of the idele class group $\mathbb{I}_{E}/E^{\ast}$ such that $\nu^2$ is trivial and $\nu$ restricted to the idele class group of $K$, $\mathbb{I}_{K}/K^{\ast}$ is also trivial? Does this imply $\nu$ is necessarily the trivial character?
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Found one class of examples while going through the list of transitive subgroups of $S_6$. Take a Galois extension say $L$ of $K$ with Galois group isomorphic to $A_4$, and $E$ be the fixed field corresponding to the Klein-4 subgroup which is even Galois over $K$, $\nu$ to be quadratic character of $E$ which corresponds to one of of the three quadratic extensions of $E$ contained in $L$. Then since the transfer homomorphism from the abelianization of $A_4$ to the Klein-4 subgroup is trivial, $\nu$ restricted to $\mathbb{I}_{K}/K^{\ast}$ is trivial.