# stronger Hilbert theorem 90

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

 $G_{\bar{K}/K}=\operatorname{Gal}(\bar{K}/K)$

be the absolute Galois group of $K$.

###### Theorem 1 (Hilbert 90).

Let $K$ be a field.

1. 1.
 $H^{1}(G_{\bar{K}/K},\bar{K}^{+})=0$
2. 2.
 $H^{1}(G_{\bar{K}/K},\bar{K}^{\ast})=0$
3. 3.

If $\operatorname{char}(K)$, the characteristic of $K$, does not divide $m$ (or $\operatorname{char}(K)=0$) then

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

where ${\mu}_{m}$ denotes the set of all $m^{th}$

## References

• 1 J.P. Serre, Galois Cohomology, Springer-Verlag, New York.
• 2 J.P. Serre, , Springer-Verlag, New York.
Title stronger Hilbert theorem 90 StrongerHilbertTheorem90 2013-03-22 13:50:27 2013-03-22 13:50:27 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 20J06 Hilbert 90 HilbertTheorem90