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stronger Hilbert theorem 90
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(Theorem)
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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.
- $$ H^1(G_{\bar{K}/K},\bar{K}^+)=0$$
- $$ H^1(G_{\bar{K}/K},\bar{K}^{\ast})=0$$
- 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 $m^{th}$ roots of unity.
- 1
- J.P. Serre, Galois Cohomology, Springer-Verlag, New York.
- 1
- J.P. Serre, Local Fields, Springer-Verlag, New York.
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"stronger Hilbert theorem 90" is owned by alozano.
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Cross-references: unity, divide, characteristic, absolute Galois group, multiplication, operation, abelian group, algebraic closure, field
This is version 3 of stronger Hilbert theorem 90, born on 2003-08-11, modified 2005-06-01.
Object id is 4577, canonical name is StrongerHilbertTheorem90.
Accessed 3566 times total.
Classification:
| AMS MSC: | 20J06 (Group theory and generalizations :: Connections with homological algebra and category theory :: Cohomology of groups) |
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Pending Errata and Addenda
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