# $[J_F : P_FN_{E/F}J_E] = 2$ for quadratic extensions of global fields of characteristic 2?

Let $E/F$ be a quadratic extension of global fields. Denote by $J_F$ the idèle group of $F$, by $P_F$ the subgroup of principal idèles and by $N_{E/F} : J_E \to J_F$ the norm map between idèles. In Timothy O'Meara's "Introduction to Quadratic Forms" it is shown that $[J_F : P_FN_{E/F}J_E] = 2$ (65:21), which is then used to prove a 'converse' to the Hilbert Reciprocity Law.

However, O'Meara derives these results under the assumption that the global characteristic is not 2. Does the statement hold if the characteristic is 2? I am interested mainly in this rather explicit converse of the Reciprocity Law and how it could be reformulated if the characteristic is 2, and the index equality seems to be the key lemma.

Yes, this is true for separable extensions of global fields of arbitrary characteristic. More generally, Artin reciprocity gives an isomorphism $J_F / P_F {N_{E/F} J_E} \cong \operatorname{Gal}(E/F)^{\text{ab}}$, and in your case the latter is order 2. (In most expositions you'll see the left hand side written as $C_F / N_{E/F} C_E$.)