= Herbrand quotient
{c}
{title2=$h_G(M)=\#\widehat H^0(G,M)/\#\widehat H^{-1}(G,M)$}
{wiki}
When the two indicated groups are finite, their size ratio is multiplicative on short exact sequences. It is one for finite modules and therefore unchanged by finite-index modifications. A trivial integral module $\mathbb Z$ has quotient $|G|$, while an induced regular lattice $\mathbb Z_p[G]$ has both Tate groups zero and quotient one. These facts permit norm-index calculations without explicitly finding every norm.
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