capitalized names (Galois, Kummer); hyphenated co-restriction (otherwise it looks like core-striction); some syntax, and spelling.

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edited Mar 14, 2017 at 23:46

David Handelman

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Is corestrictionco-restriction in Galois cohomology in fact the norm map via kummerKummer isomorphism?

Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ such that satisfies $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the absolute galoisGalois group of $\mathrm{F}$, denoted by $G_F$, and its open sub-groupsubgroup $G_{F[\sqrt[p]{a}]}$ of index $p$ which(which is the absolute galoisGalois group of the extension $F[\sqrt[p]{a}]$). There is a well-known map between the following galoisGalois cohomology groups, which is the corestrictionco-restriction: $$Cor:H^1(G_{F[\sqrt[p]{a}]}, \mathbb{Z}/p) \rightarrow H^1(G_F, \mathbb{Z}/p)$$ In addition, there is a map between the abelian groups $F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p$ and $F^*/{F^*}^p$ which is the field norm: $$ N_{F[\sqrt[p]{a}] / F} : F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p \rightarrow F^*/{F^*}^p $$ Kummer theory yields the isomorphisms $H^1(G_F, \mathbb{Z}/p) \cong F^*/{F^*}^p$ and $H^1(G_{F[\sqrt[p]{a}]}, \mathbb{Z}/p) \cong F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p$, so the natural question to be asked is wheterwhether the corestrictionco-restriction map and the norm map are actually the same thing, once looking onvia these identifications? In other words, does the corresponding diagram commutescommute? I couldn't find any referencereferences for thatthis.

Is corestriction in Galois cohomology in fact the norm map via kummer isomorphism?

Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ that satisfies $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the absolute galois group of $\mathrm{F}$, denoted by $G_F$, and its open sub-group $G_{F[\sqrt[p]{a}]}$ of index $p$ which is the absolute galois group of the extension $F[\sqrt[p]{a}]$. There is a well-known map between the following galois cohomology groups, which is the corestriction: $$Cor:H^1(G_{F[\sqrt[p]{a}]}, \mathbb{Z}/p) \rightarrow H^1(G_F, \mathbb{Z}/p)$$ In addition, there is a map between the abelian groups $F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p$ and $F^*/{F^*}^p$ which is the field norm: $$ N_{F[\sqrt[p]{a}] / F} : F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p \rightarrow F^*/{F^*}^p $$ Kummer theory yields the isomorphisms $H^1(G_F, \mathbb{Z}/p) \cong F^*/{F^*}^p$ and $H^1(G_{F[\sqrt[p]{a}]}, \mathbb{Z}/p) \cong F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p$, so the natural question to be asked is wheter the corestriction map and the norm map are actually the same thing, once looking on these identifications? In other words, does the corresponding diagram commutes? I couldn't find any reference for that.

Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism?

Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ such that $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the absolute Galois group of $\mathrm{F}$, denoted by $G_F$, and its open subgroup $G_{F[\sqrt[p]{a}]}$ of index $p$ (which is the absolute Galois group of the extension $F[\sqrt[p]{a}]$). There is a well-known map between the following Galois cohomology groups, the co-restriction: $$Cor:H^1(G_{F[\sqrt[p]{a}]}, \mathbb{Z}/p) \rightarrow H^1(G_F, \mathbb{Z}/p)$$ In addition, there is a map between the abelian groups $F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p$ and $F^*/{F^*}^p$ which is the field norm: $$ N_{F[\sqrt[p]{a}] / F} : F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p \rightarrow F^*/{F^*}^p $$ Kummer theory yields the isomorphisms $H^1(G_F, \mathbb{Z}/p) \cong F^*/{F^*}^p$ and $H^1(G_{F[\sqrt[p]{a}]}, \mathbb{Z}/p) \cong F[\sqrt[p]{a}]^*/{F[\sqrt[p]{a}]^*}^p$, so the natural question to be asked is whether the co-restriction map and the norm map are actually the same thing, via these identifications? In other words, does the corresponding diagram commute? I couldn't find any references for this.