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Keenan Kidwell
Keenan Kidwell
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Well, youYou have the inflation restriction sequence $0\rightarrow H^1(K^{nr}/K,\mathbf{F}_p)\rightarrow H^1(K,\mathbf{F}_p)\rightarrow H^1(K^{nr},\mathbf{F}_p)^{\mathrm{Gal}(K^{nr}/K)}\rightarrow 0$ with the zero on the end because $\hat{\mathbf{Z}}$ has cohomological dimension one. The first term is $\mathbf{F}_p$ by evaluation on Frobenius. The second term can be understood via class field theory. Any continuous homomorphism $G_K\rightarrow\mathbf{F}_p$ factors through the maximal abelian pro-$p$ quotient of $G_K$, which is isomorphic to $\mathbf{Z}_p^{d+1}\times\mu$ where $d$ is the degree over $\mathbf{Q}_p$ and $\mu$ is the group of $p$-power roots of unity in $K$. Since $\mathrm{Hom}_{cts}(\mathbf{Z}_p^{d+1},\mathbf{F}_p)$ is just $d+1$ copies of $\mathrm{Hom}(\mathbf{F}_p,\mathbf{F}_p)$, its dimension is $d+1$. If $\mu$ is trivial, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=0$. If $K$ contains a primitive $p$-th root of unity, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=\mathrm{Hom}(\mu_p,\mathbf{F}_p)$ is $1$-dimensional. So the dimension of the middle piece is $d+1$ or $d+2$ according as $K$ does or doesn't contain a primitive $p$-th root of unity. So the dimension of the last term is either $d$ or $d+1$.

Well, you have the inflation restriction sequence $0\rightarrow H^1(K^{nr}/K,\mathbf{F}_p)\rightarrow H^1(K,\mathbf{F}_p)\rightarrow H^1(K^{nr},\mathbf{F}_p)^{\mathrm{Gal}(K^{nr}/K)}\rightarrow 0$ with the zero on the end because $\hat{\mathbf{Z}}$ has cohomological dimension one. The first term is $\mathbf{F}_p$ by evaluation on Frobenius. The second term can be understood via class field theory. Any continuous homomorphism $G_K\rightarrow\mathbf{F}_p$ factors through the maximal abelian pro-$p$ quotient of $G_K$, which is isomorphic to $\mathbf{Z}_p^{d+1}\times\mu$ where $d$ is the degree over $\mathbf{Q}_p$ and $\mu$ is the group of $p$-power roots of unity in $K$. Since $\mathrm{Hom}_{cts}(\mathbf{Z}_p^{d+1},\mathbf{F}_p)$ is just $d+1$ copies of $\mathrm{Hom}(\mathbf{F}_p,\mathbf{F}_p)$, its dimension is $d+1$. If $\mu$ is trivial, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=0$. If $K$ contains a primitive $p$-th root of unity, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=\mathrm{Hom}(\mu_p,\mathbf{F}_p)$ is $1$-dimensional. So the dimension of the middle piece is $d+1$ or $d+2$ according as $K$ does or doesn't contain a primitive $p$-th root of unity. So the dimension of the last term is either $d$ or $d+1$.

You have the inflation restriction sequence $0\rightarrow H^1(K^{nr}/K,\mathbf{F}_p)\rightarrow H^1(K,\mathbf{F}_p)\rightarrow H^1(K^{nr},\mathbf{F}_p)^{\mathrm{Gal}(K^{nr}/K)}\rightarrow 0$ with the zero on the end because $\hat{\mathbf{Z}}$ has cohomological dimension one. The first term is $\mathbf{F}_p$ by evaluation on Frobenius. The second term can be understood via class field theory. Any continuous homomorphism $G_K\rightarrow\mathbf{F}_p$ factors through the maximal abelian pro-$p$ quotient of $G_K$, which is isomorphic to $\mathbf{Z}_p^{d+1}\times\mu$ where $d$ is the degree over $\mathbf{Q}_p$ and $\mu$ is the group of $p$-power roots of unity in $K$. Since $\mathrm{Hom}_{cts}(\mathbf{Z}_p^{d+1},\mathbf{F}_p)$ is just $d+1$ copies of $\mathrm{Hom}(\mathbf{F}_p,\mathbf{F}_p)$, its dimension is $d+1$. If $\mu$ is trivial, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=0$. If $K$ contains a primitive $p$-th root of unity, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=\mathrm{Hom}(\mu_p,\mathbf{F}_p)$ is $1$-dimensional. So the dimension of the middle piece is $d+1$ or $d+2$ according as $K$ does or doesn't contain a primitive $p$-th root of unity. So the dimension of the last term is either $d$ or $d+1$.

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Keenan Kidwell
Keenan Kidwell
  • 3.4k
  • 1
  • 21
  • 35

Well, you have the inflation restriction sequence $0\rightarrow H^1(K^{nr}/K,\mathbf{F}_p)\rightarrow H^1(K,\mathbf{F}_p)\rightarrow H^1(K^{nr},\mathbf{F}_p)^{\mathrm{Gal}(K^{nr}/K)}\rightarrow 0$ with the zero on the end because $\hat{\mathbf{Z}}$ has cohomological dimension one. The first term is $\mathbf{F}_p$ by evaluation on Frobenius. The second term can be understood via class field theory. Any continuous homomorphism $G_K\rightarrow\mathbf{F}_p$ factors through the maximal abelian pro-$p$ quotient of $G_K$, which is isomorphic to $\mathbf{Z}_p^{d+1}\times\mu$ where $d$ is the degree over $\mathbf{Q}_p$ and $\mu$ is the group of $p$-power roots of unity in $K$. Since $\mathrm{Hom}_{cts}(\mathbf{Z}_p^{d+1},\mathbf{F}_p)$ is just $d+1$ copies of $\mathrm{Hom}(\mathbf{F}_p,\mathbf{F}_p)$, its dimension is $d+1$. If $\mu$ is trivial, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=0$. If $K$ contains a primitive $p$-th root of unity, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=\mathrm{Hom}(\mu_p,\mathbf{F}_p)$ is $1$-dimensional. So the dimension of the middle piece is $d+1$ or $d+2$ according as $K$ does or doesn't contain a primitive $p$-th root of unity. So the dimension of the last term is either $d$ or $d+1$.