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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
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References:

[1] Aljadeff, E., Sonn, J., Relative Brauer groups and m-torsion. Proc. Amer. Math. Soc.130 (2002), 1333-1337. · Zbl 1099.11066
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