  In this paper we study the relationship between degeneracy and
decomposability in abelian crossed products. In particular we construct an
indecomposable abelian crossed product division algebra of exponent $p$ and
index $p^2$ for $p$ an odd prime. The algebra we construct is generic in the
sense of Amitsur and Saltman 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
[Karpenko, Codimension 2 cycles on Severi-Brauer varieites (1998)] and
[McKinnie, Indecomposable $p$-algebras and Galois subfields in generic abelian
crossed products (2008)]. 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.
