This is a question I asked on math.stackexchange without success.


Let $p$ be a prime number and denote by $\mathbb{F}\_p(1)$ the one dimensional vector space over $\mathbb{F}\_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}\_p / \mathbb{Q}\_p)$ via the mod $p$ cyclotomic character.


We have a nice description of the vector space $H^1(G, \mathbb{F}\_p(1))$ given by the Kummer isomorphism $H^1(G, \mathbb{F}\_p(1)) \simeq \mathbb{Q}\_p^\* / \left(\mathbb{Q}\_p^\*\right)^p$.


Is there also a nice description of $H^1(G, \mathbb{F}\_p(n))$ where $n$ is any (positive) integer? (here $\mathbb{F}\_p(n)$ is just $\mathbb{F}\_p$ with the action of the $n$-th power of the mod $p$ cyclotomic character)



Generalization of Kummer isomorphism?
It depends what you mean by "nice". There is, to the best of my knowledge, no such easy explicit description of $H^1(G,F\_p(n))$ if $n \neq 0,1$. If $n=1$, it's Kummer theory as you have described. If $n=0$, the $H^1$ is the group of homomorphisms of $G$ into $F\_p$, so it boils down to local class field theory.


In general, you can compute the dimension of $H^1(G,F\_p(n))$ using the argument suggested by Timo Keller.


If you were looking at $H^1(G,Q\_p(n))$ instead of $H^1(G,F\_p(n))$, then you'd still have the "explicit" description as above, but in addition for $n \geq 2$, you'd have the Bloch-Kato exponential map, $exp\_{Q\_p(n)}$ which is (for $n \geq 2$) an isomorphism between a certain $1$-dimensional vector space $D\_{cris}(Q\_p(n))$ and $H^1(G,Q\_p(n))$. This map is however not naturally "integral". This is typical of $p$-adic Hodge theory: as the "weight" increases, you have to replace simple integrality properties by more complicated $p$-adic analytical properties.


Another direction would be to use the theory of $(\phi,\Gamma)$-modules. The elements of $H^1(G,F\_p(n))$ can then be seen as pairs of elements of $F\_p[[X]][1/X]$ satisfying certain properties.


