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

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

(Here p$p$ is a prime and k$k$ is a local field or global field of finite type, i.e finite extension of Q$\mathbb{Q}$ of Qp$\mathbb{Q}_p$.)

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

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

This This induces a homomorphism from H^2(G(K/k),F_p)$H^2(G(K/k),\mathbb{F}_p)$ to H^2(G(K'/k'),F_p)$H^2(G(K'/k'),\mathbb{F}_p)$.

The question is, Isis this map injective?

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?

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 $\mathbb{Q}$ of $\mathbb{Q}_p$.)

As $G(K/k)$ is a pro-p group, we study $H^2(G(K/k), \mathbb{F}_p)$ where $\mathbb{F}_p$ is the finite field with $p$ elements with trivial group action.

Let $k'$ be the field generated by $k$ and the $p$'th 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 $H^2(G(K/k),\mathbb{F}_p)$ to $H^2(G(K'/k'),\mathbb{F}_p)$.

The question is, is this map injective?

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gualterio
gualterio
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gualterio
gualterio
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A question on the injectivity of a canonical map between galois cohomology groups

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?