Dec groups for arbitrarily high exponents (Q1318194)

Let \(K/F\) be an abelian Galois extension of fields. The group \(\text{Dec}(K/F)\) is the subgroup of the relative Brauer group \(\text{Br}(K/F)\) generated by the relative Brauer groups \(\text{Br}(L/F)\) of all the cyclic extensions \(L/F\) contained in \(K\). This group was introduced by the reviewer [J. Algebra 70, 420-436 (1980; Zbl 0473.16004)] in relation with the construction of indecomposable division algebras of prime exponent. If \(K/F\) is elementary abelian of degree 4, then it is known that \(\text{Dec}(K/F)\) is the 2-torsion subgroup in \(\text{Br}(K/F)\). In the present paper, the author explicitly constructs for each prime \(p\) and each integer \(n\geq 1\) (\(n \geq 2\) if \(p = 2\)) a field \(F\), an abelian extension \(K/F\) with Galois group \(({\mathbb{Z}}/p^ n \mathbb{Z})\times (\mathbb{Z}/p\mathbb{Z})\) and a central simple algebra \(A\) of exponent \(p\) split by \(K\) whose Brauer class is not in \(\text{Dec}(K/F)\). The base field \(F\) is a rational function field in three variables over a field of characteristic zero containing sufficiently many roots of unity. The methods of proof are essentially valuation-theoretic.