$H^i(G, \mathcal{O}_L^\times)$.

In this paper we provide both widely-applicable estimates for $m(L/K)$ and exact formulas in several special cases. Our first result is the following general upper bound for $m(L/K)$, that will be proved in Subsection \[sect:GeneralBound\]. Notice that $m(L/K)$ is by definition a power of $p$, so bounding its absolute value is equivalent to bounding its $p$-adic valuation.

\[thm:IntroGeneralBound\] Let $L/K$ be a Galois extension of $p$-adic fields. Then $$v_p (m(L/K)) {\leqslant}f_L(e_{L/K}-1) + \frac{1}{2} [L:\mathbb{Q}_p]  \cdot v_p([L:K]).$$

We note that the bound of the previous theorem can be relaxed to an estimate depending only on $[L:\mathbb{Q}_p]$.

Much more is known on the Galois structure of $L/K$ when $L$ is absolutely abelian. As a consequence, in this case we obtain an explicit formula for the value of $m(L/K)$.

\[thm:IntroAbsolutelyAbelian\] Let $L/K$ be a Galois extension of $p$-adic fields, with $p$ odd, and assume that $L/{\mathbb{Q}}_p$ is abelian. Denote by $L^{nr}$ the maximal unramified subextension of $L/K$. Then $$\begin{array}{rl}
 v_p( m(L/K))&=v_{p}(m(L/L^{nr}))\\
  &  \displaystyle={\frac{f_L}{2}\left(e_{L}\cdot v_p(e_{L/K})- \sum_{d \mid e_{L/K}} \frac{\varphi(d)}{[L^{nr}(\zeta_d):L^{nr}]} v_{L^{nr}}( \operatorname{disc}(L^{nr}(\zeta_d)/L^{nr}))\right)}.
\end{array}$$

We prove Theorem \[thm:IntroAbsolutelyAbelian\] in Section \[sec:assab\].

In Section \[sec:local-global\] we consider an analogous quantity $m(E/F)$ for a number field extension $E/F$ and, in the case when $F=\mathbb{Q}$ and $E/\mathbb{Q}$ 