= Hilbert's theorem 90
{c}
{title2=$H^1(K,\overline K^*)=0$}
{wiki}
= Hilbert theorem 90
{c}
{synonym}
For a finite Galois extension $L/K$, multiplicative Hilbert theorem 90 says $H^1(\operatorname{Gal}(L/K),L^*)=0$. A multiplicative cocycle has the form $c_\sigma=\sigma(b)/b$. One proof forms a nonzero sum $A=\sum_\sigma c_\sigma\sigma(a)$, using linear independence of field automorphisms. Its transformation law is $\tau A=c_\tau^{-1}A$, so $b=A^{-1}$ gives $c_\tau=\tau(b)/b$. Passing to the algebraic closure gives $H^1(K,\overline K^*)=0$, so the multiplicative Kummer sequence identifies $H^1(K,\mu_m)$ with $K^*/K^{*m}$.
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