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In Springer's Encyclopaedia of Mathematics> Galois Cohomology, it is mentioned that

For an imperfect field $k$, $H^1(k,\mathbb{G}_a)\neq 0$ in general.

I'm looking for such an example or a reference to one.

(Here $\mathbb{G}_a$ is the additive group of a field and $H^1(k,\mathbb{G}_a)$ means the first cohomology of the Galois group of $K/k$ with values in $\mathbb{G}_a$, where $K$ is the separable algebraic closure of $k$)

Many Thanks!

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I don't often think about imperfect fields, but I think that the cited claim is false. Indeed, first reduce to finite Galois extensions: if $G$ is the absolute Galois group of $k$ and $M$ is any $G$-module, then $ H^r(G,M) = \varinjlim H^r(G/H,M^H), $ where the limit is over the open normal subgroups of $G$. (The isomorphism from right to left is given by the obvious inflation maps, and the cohomology on the LHS is of course the continuous cohomology.)

Now, let $L/k$ be a finite Galois extension with Galois group $\Gamma$. By the normal basis theorem, the additive group $L$ is isomorphic to $k[\Gamma]\cong \rm{Ind}_{G/\{1\}} k$ as Galois modules. By Shapiro's Lemma, we therefore have $H^r(\Gamma,L) = H^r(\{1\},k) = 0$ for any $r>0$.