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)