I'm currently reading the book "Galois theory of p-extensions" by Helmut Koch.

There, we calculate the cohomological dimension of the galois group G(K/k) where K is the maximal (normal) p-extension of k.

(Here p is a prime and k is a local field or global field of finite type, i.e finite extension of Q of Qp)

As G(K/k) is a pro-p group, we study H^2(G(K/k), F_p) where F_p is the finite field with p elements with trivial group action.

Let k' be the field generated by k and the pth roots of unity. And let K' be the maximal p-extension of k'. Then there is a canonical group homomorphism from G(K'/k') to G(K/k) (restriction map).

This induces a homomorphism from H^2(G(K/k),F_p) to H^2(G(K'/k'),F_p).

The question is, Is this map injective?