A Galois character is a continuous one-dimensional Galois representation: the absolute Galois group acts by multiplication by a scalar in a coefficient field . A quadratic character over a coefficient field of characteristic different from two has values in ; after identifying this group with , its kernel fixes a quadratic extension, or the base field for the trivial character. Its restriction to an inertia group is trivial exactly when that extension is unramified at the corresponding place.
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