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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?

rtybase
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1 Answers1

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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

nguyen quang do
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  • In summary, after a counter-example due to Ax, we know that trivial Brauer group does not imply $C_1$. In fact, there exists a field satisfying even a stronger variant namely dim $\leq 1$ (meaning all algebraic extensions also have Brauer group trivial) which is not $C_r$ for $r>0$. – mathemather Aug 15 '23 at 11:57