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I have a question about Chapter XIV of "Corps Locaux" by Serre.

Sketch of the situation: let $K$ be a field with separable closure $\overline{K}$. Let $\Gamma_K$ be the absolute Galois group. The group of characters $H^1(K,\mathbb{Q}/\mathbb{Z})$ can be identified with $H^2(K,\mathbb{Z})$. Cup-product then gives a pairing $K^\times \times H^1(K,\mathbb{Q}/\mathbb{Z}) \to H^2(K,\overline{K}^\times) = \text{Br}(K)$. For every $\alpha \in K^\times$ and $\chi \in H^1(K,\mathbb{Q}/\mathbb{Z})$ this defines an element of $\text{Br}(K)$ which we simply denote by $(\alpha,\chi)$.

Now the exercise on page 205 (English translation) gives the following rules to calculate the corestriction from a finite extension $L \subseteq \overline{K}$ to $K$ of such an element $(\alpha,\chi)$:

$\text{Cores}_{L/K}(\alpha,\chi) = (\alpha,\text{Cores}_{L/K}(\chi))$ if $\alpha \in K^\times$ and $\chi \in H^1(L,\mathbb{Q}/\mathbb{Z})$

$\text{Cores}_{L/K}(\alpha,\chi) = (N_{L/K}(\alpha),\chi)$ if $\alpha \in L^\times$ and $\chi \in H^1(K,\mathbb{Q}/\mathbb{Z})$

However, I would like to know how to calculate

$\text{Cores}_{L/K}(\alpha,\chi)$ for $\alpha \in L^\times$ and $\chi \in H^1(L,\mathbb{Q}/\mathbb{Z})$

and it is not clear to me what the formulas should become. Any help would be very welcome!

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I know nothing about corestrictions in cohomology, but only possess a limited amount of knowledge about corestriction of algebras. Perhaps they are related? And I understand not the difficulties: are you asking how to expand this definition to the whole of L and $H^1(L,Q/Z)$? Thanks in any case for explaining... –  awllower Jun 8 '12 at 16:07
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