Field with vanishing Brauer group which is not $C_1$
In Serre's Local Fields he gives several examples of fields with trivial Brauer group. However, all of these examples are $C_1$ or conjectured to be $C_1$.
Is there an example of a field which is not $C_1$ but has a trivial Brauer group?
algebraic-number-theory brauer-group
edited 12 hours ago
rtybase
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In Serre's Local Fields he gives several examples of fields with trivial Brauer group. However, all of these examples are $C_1$ or conjectured to be $C_1$.
Is there an example of a field which is not $C_1$ but has a trivial Brauer group?
algebraic-number-theory brauer-group
edited 12 hours ago
rtybase
10.4k21433
asked yesterday
1213
1767
add a comment |
In Serre's Local Fields he gives several examples of fields with trivial Brauer group. However, all of these examples are $C_1$ or conjectured to be $C_1$.
Is there an example of a field which is not $C_1$ but has a trivial Brauer group?
algebraic-number-theory brauer-group
edited 12 hours ago
rtybase
10.4k21433
asked yesterday
1213
1767
In Serre's Local Fields he gives several examples of fields with trivial Brauer group. However, all of these examples are $C_1$ or conjectured to be $C_1$.
Is there an example of a field which is not $C_1$ but has a trivial Brauer group?
algebraic-number-theory brauer-group
algebraic-number-theory brauer-group
edited 12 hours ago
rtybase
10.4k21433
asked yesterday
1213
1767
edited 12 hours ago
rtybase
10.4k21433
asked yesterday
1213
1767
edited 12 hours ago
rtybase
10.4k21433
edited 12 hours ago
rtybase
10.4k21433
edited 12 hours ago
rtybase
10.4k21433
10.4k21433
asked yesterday
1213
1767
asked yesterday
1213
1767
asked yesterday
1213
1767
1767
add a comment |
add a comment |
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In Serre's "Galois Cohomology" II-3 , a field $k$ is said to have "dimension $1$", $dim(k)le 1$ for short, iff $Br(K)=0$ for all algebraic extensions $K/k$. If moreover $k$ is perfect, then $dim(k)le 1$ iff $k$ has cohomological dimension $1$ (i.e. $cd(G_k)le 1$). Besides, if $k$ is $C_1$, then $dim(k)le 1$ and $[k:k^p]=1$ or $p$ @; this implies that a perfect $k$ which is $C_1$ has dimension $1$. Serre also asked (rather dubiously) whether the two properties stated in @ were equivalent, but soon after a counter-example was given by Ax (1965), who even constructed a field $k$ of dimension $1$ which is not $C_r$ for all $r$ (see the reference in [CT]).
More recently (2005), Colliot-Thélène [CT] came back to the question "coh. dim. $1$ vs $C_1$", and he showed in particular, starting with a field of characteristic $neq p$ and using Severi-Brauer varieties, how to construct systematically an extension $F/k$ s.t. $G_F$ is a pro-$p$-group and $cd(G_F)le 1$.
[CT] J.-L. Colliot-thélène, Fields of cohomological dimension $1$ versus $C_1$-fields, arXiv:math/0502194
answered 13 hours ago
nguyen quang do
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In Serre's "Galois Cohomology" II-3 , a field $k$ is said to have "dimension $1$", $dim(k)le 1$ for short, iff $Br(K)=0$ for all algebraic extensions $K/k$. If moreover $k$ is perfect, then $dim(k)le 1$ iff $k$ has cohomological dimension $1$ (i.e. $cd(G_k)le 1$). Besides, if $k$ is $C_1$, then $dim(k)le 1$ and $[k:k^p]=1$ or $p$ @; this implies that a perfect $k$ which is $C_1$ has dimension $1$. Serre also asked (rather dubiously) whether the two properties stated in @ were equivalent, but soon after a counter-example was given by Ax (1965), who even constructed a field $k$ of dimension $1$ which is not $C_r$ for all $r$ (see the reference in [CT]).
More recently (2005), Colliot-Thélène [CT] came back to the question "coh. dim. $1$ vs $C_1$", and he showed in particular, starting with a field of characteristic $neq p$ and using Severi-Brauer varieties, how to construct systematically an extension $F/k$ s.t. $G_F$ is a pro-$p$-group and $cd(G_F)le 1$.
[CT] J.-L. Colliot-thélène, Fields of cohomological dimension $1$ versus $C_1$-fields, arXiv:math/0502194
answered 13 hours ago
nguyen quang do
8,4111723
add a comment |
In Serre's "Galois Cohomology" II-3 , a field $k$ is said to have "dimension $1$", $dim(k)le 1$ for short, iff $Br(K)=0$ for all algebraic extensions $K/k$. If moreover $k$ is perfect, then $dim(k)le 1$ iff $k$ has cohomological dimension $1$ (i.e. $cd(G_k)le 1$). Besides, if $k$ is $C_1$, then $dim(k)le 1$ and $[k:k^p]=1$ or $p$ @; this implies that a perfect $k$ which is $C_1$ has dimension $1$. Serre also asked (rather dubiously) whether the two properties stated in @ were equivalent, but soon after a counter-example was given by Ax (1965), who even constructed a field $k$ of dimension $1$ which is not $C_r$ for all $r$ (see the reference in [CT]).
More recently (2005), Colliot-Thélène [CT] came back to the question "coh. dim. $1$ vs $C_1$", and he showed in particular, starting with a field of characteristic $neq p$ and using Severi-Brauer varieties, how to construct systematically an extension $F/k$ s.t. $G_F$ is a pro-$p$-group and $cd(G_F)le 1$.
[CT] J.-L. Colliot-thélène, Fields of cohomological dimension $1$ versus $C_1$-fields, arXiv:math/0502194
answered 13 hours ago
nguyen quang do
8,4111723
add a comment |
In Serre's "Galois Cohomology" II-3 , a field $k$ is said to have "dimension $1$", $dim(k)le 1$ for short, iff $Br(K)=0$ for all algebraic extensions $K/k$. If moreover $k$ is perfect, then $dim(k)le 1$ iff $k$ has cohomological dimension $1$ (i.e. $cd(G_k)le 1$). Besides, if $k$ is $C_1$, then $dim(k)le 1$ and $[k:k^p]=1$ or $p$ @; this implies that a perfect $k$ which is $C_1$ has dimension $1$. Serre also asked (rather dubiously) whether the two properties stated in @ were equivalent, but soon after a counter-example was given by Ax (1965), who even constructed a field $k$ of dimension $1$ which is not $C_r$ for all $r$ (see the reference in [CT]).
More recently (2005), Colliot-Thélène [CT] came back to the question "coh. dim. $1$ vs $C_1$", and he showed in particular, starting with a field of characteristic $neq p$ and using Severi-Brauer varieties, how to construct systematically an extension $F/k$ s.t. $G_F$ is a pro-$p$-group and $cd(G_F)le 1$.
[CT] J.-L. Colliot-thélène, Fields of cohomological dimension $1$ versus $C_1$-fields, arXiv:math/0502194
answered 13 hours ago
nguyen quang do
8,4111723
In Serre's "Galois Cohomology" II-3 , a field $k$ is said to have "dimension $1$", $dim(k)le 1$ for short, iff $Br(K)=0$ for all algebraic extensions $K/k$. If moreover $k$ is perfect, then $dim(k)le 1$ iff $k$ has cohomological dimension $1$ (i.e. $cd(G_k)le 1$). Besides, if $k$ is $C_1$, then $dim(k)le 1$ and $[k:k^p]=1$ or $p$ @; this implies that a perfect $k$ which is $C_1$ has dimension $1$. Serre also asked (rather dubiously) whether the two properties stated in @ were equivalent, but soon after a counter-example was given by Ax (1965), who even constructed a field $k$ of dimension $1$ which is not $C_r$ for all $r$ (see the reference in [CT]).
More recently (2005), Colliot-Thélène [CT] came back to the question "coh. dim. $1$ vs $C_1$", and he showed in particular, starting with a field of characteristic $neq p$ and using Severi-Brauer varieties, how to construct systematically an extension $F/k$ s.t. $G_F$ is a pro-$p$-group and $cd(G_F)le 1$.
[CT] J.-L. Colliot-thélène, Fields of cohomological dimension $1$ versus $C_1$-fields, arXiv:math/0502194
answered 13 hours ago
nguyen quang do
8,4111723
answered 13 hours ago
nguyen quang do
8,4111723
answered 13 hours ago
nguyen quang do
8,4111723
answered 13 hours ago
nguyen quang do
8,4111723
8,4111723
add a comment |
add a comment |
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