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 $(\overline{K},+)$ and similarly ${\overline{K}}^{\ast }=(\overline{K},\ast )$ (here the operation is multiplication). Also we let

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

be the absolute Galois 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 $K$, does not 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.

Title	stronger Hilbert theorem 90
Canonical name	StrongerHilbertTheorem90
Date of creation	2013-03-22 13:50:27
Last modified on	2013-03-22 13:50:27
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Theorem
Classification	msc 20J06
Synonym	Hilbert 90
Related topic	HilbertTheorem90