stronger Hilbert theorem 90

Let $K$ be a field and let $\overline{K}$ be an algebraic closure of$K$. By ${\overline{K}}^{+}$ we denote the abelian group $(\overline{K},+)$ andsimilarly ${\overline{K}}^{\ast }=(\overline{K},\ast )$ (here the operation ismultiplication). Also we let

$$G_{\overline{K}/K}=\mathrm{Gal}(\overline{K}/K)$$

be the absoluteGalois group of $K$.

Theorem 1 (Hilbert 90).

Let $K$ be a field.

1. $$H^1(G_{\overline{K}/K},{\overline{K}}^{+})=0$$

2. $$H^1(G_{\overline{K}/K},{\overline{K}}^{\ast })=0$$

3. If $\mathrm{char}(K)$, the characteristic of $K$, doesnot divide $m$ (or $\mathrm{char}(K)=0$) then

   $$H^1(G_{\overline{K}/K},{\mu }_m)\cong K^{\ast }/K^{\ast m}$$

   where ${\mu }_m$ denotes the set of all $m^{th}$-roots of unity.