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It is perhaps helpful to look at the following papers of Fein and Schacher:

Fein; Schacher; Brauer groups of fields algebraic over Q. J. Algebra 43 (1976), no. 1, 328–337.

Fein; Schacher; Divisible groups that are Brauer groups. Commun. Algebra 7, 989-994 (1979).

I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group $G$ is the Brauer group of some field $K$ if and only if $G$ is the direct sum of a divisible group and a torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).

Also the authors of the following paper

Auel; Brussel; Garibaldi; Vishne; Open problems on central simple algebras. Transform. Groups 16 (2011), no. 1, 219–264.

cite a Russian paper of Merkurjev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.

It is perhaps helpful to look at the following papers of Fein and Schacher:

Fein; Schacher; Brauer groups of fields algebraic over Q. J. Algebra 43 (1976), no. 1, 328–337.

Fein; Schacher; Divisible groups that are Brauer groups. Commun. Algebra 7, 989-994 (1979).

I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group $G$ is the Brauer group of some field $K$ if and only if $G$ is the direct sum of a divisible group and a $2$-torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).

Also the authors of the following paper

Auel; Brussel; Garibaldi; Vishne; Open problems on central simple algebras. Transform. Groups 16 (2011), no. 1, 219–264.

cite a Russian paper of Merkurjev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.

It is perhaps helpful to look at the following papers of Fein and Schacher:

Fein; Schacher; Brauer groups of fields algebraic over Q. J. Algebra 43 (1976), no. 1, 328–337.

Fein; Schacher; Divisible groups that are Brauer groups. Commun. Algebra 7, 989-994 (1979).

I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group $G$ is the Brauer group of some field $K$ if and only if $G$ is the direct sum of a divisible group and a torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).

Also the authors of the following paper

Auel; Brussel; Garibaldi; Vishne; Open problems on central simple algebras. Transform. Groups 16 (2011), no. 1, 219–264.

cite a Russian paper of Merkuriev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.

It is perhaps helpful to look at the following papers of Fein and Schacher:

Fein; Schacher; Brauer groups of fields algebraic over Q. J. Algebra 43 (1976), no. 1, 328–337.

Fein; Schacher; Divisible groups that are Brauer groups. Commun. Algebra 7, 989-994 (1979).

I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group $G$ is the Brauer group of some field $K$ if and only if $G$ is the direct sum of a divisible group and a torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).

Also the authors of the following paper

Auel; Brussel; Garibaldi; Vishne; Open problems on central simple algebras. Transform. Groups 16 (2011), no. 1, 219–264.

cite a Russian paper of Merkurjev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.

It is perhaps helpful to look at the following papers of Fein and Schacher:

Fein; Schacher; Brauer groups of fields algebraic over Q. J. Algebra 43 (1976), no. 1, 328–337.

Fein; Schacher; Divisible groups that are Brauer groups. Commun. Algebra 7, 989-994 (1979).

I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group is the Brauer group of some field of $K$ if and only if $G$ is the direct sum of a divisible group and a torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).

Also the authors of the following paper

Auel; Brussel; Garibaldi; Vishne; Open problems on central simple algebras. Transform. Groups 16 (2011), no. 1, 219–264.

cite a Russian paper of Merkuriev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.

It is perhaps helpful to look at the following papers of Fein and Schacher:

Fein; Schacher; Brauer groups of fields algebraic over Q. J. Algebra 43 (1976), no. 1, 328–337.

Fein; Schacher; Divisible groups that are Brauer groups. Commun. Algebra 7, 989-994 (1979).

I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group $G$ is the Brauer group of some field $K$ if and only if $G$ is the direct sum of a divisible group and a torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).

Also the authors of the following paper

Auel; Brussel; Garibaldi; Vishne; Open problems on central simple algebras. Transform. Groups 16 (2011), no. 1, 219–264.

cite a Russian paper of Merkuriev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.