= Galois character
{c}
{title2=$\chi:G_K\to F^\times$}
A Galois character is a continuous one-dimensional <Galois representation>: the <absolute Galois group> acts by multiplication by a scalar in a coefficient <field> $F$. A quadratic character over a coefficient field of characteristic different from two has values in $\{1,-1\}$; after identifying this group with $\mathbb Z/2\mathbb Z$, 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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