## On the $$n$$-torsion subgroup of the Brauer group of a number field.(English)Zbl 1048.11089

Let $$K$$ be an algebraic number field Galois over $$\mathbb Q$$, let $$\text{Br}(K)$$ be the Brauer group, and let $$\text{Br}_n(K)$$ be its $$n$$-torsion subgroup. The authors show that, providing $$n$$ is prime to the class number of $$K$$, then $$\text{Br}_n(K)\cong \text{Br}(L/K)$$ is an algebraic relative Brauer group for an abelian extension $$L/K$$. The proof depends on the following construction: given a prime number $$\ell$$ not dividing the class number of $$K$$ and a positive integer $$r$$, there is an abelian extension $$L/K$$ of exponent $$\ell^r$$ with local degree equal to $$\ell^r$$ at every finite prime. If $$\ell=2$$, $$L$$ can be taken to be totally complex.

### MSC:

 11R34 Galois cohomology 16K50 Brauer groups (algebraic aspects) 11R52 Quaternion and other division algebras: arithmetic, zeta functions
Full Text:

### References:

 [1] Aljadeff, E., Sonn, J., Relative Brauer groups and m-torsion. Proc. Amer. Math. Soc.130 (2002), 1333-1337. · Zbl 1099.11066 [2] Fein, B., Schacher, M., Relative Brauer groups I. J. Reine Angew. Math.321 (1981), 179-194. · Zbl 0436.13003 [3] Fein, B., Kantor, W., Schacher, M., Relative Brauer groups II. J. Reine Angew. Math.328 (1981), 39-57. · Zbl 0457.13004 [4] Fein, B., Schacher, M., Relative Brauer groups III. J. Reine Angew. Math.335 (1982), 37-39. · Zbl 0484.13005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.