Degeneracy and decomposability in abelian crossed products

In this paper we continue the study of the relationship between degeneracy and decomposability in abelian crossed products (K. McKinnie (2008) [McK08], ). In particular we construct an indecomposable abelian crossed product division algebra of exponent p and index p2 for p an odd prime. The algebra we construct is generic in the sense of S.A. Amitsur and D. Saltman (1978) [AS78], and has the property that its underlying abelian crossed product is a decomposable division algebra defined by a non-degenerate matrix. This algebra gives an example of an indecomposable generic abelian crossed product which is shown to be indecomposable without using torsion in the Chow group of the corresponding Severi–Brauer variety as was needed in N.A. Karpenko (1998) [Kar98], and K. McKinnie (2008) [McK08]. It also gives an example of a Brauer class which is in Tignol's Dec group with respect to one abelian maximal subfield, but not in the Dec group with respect to another.