A non-extended hermitian form over ℤ[ℤ]

We describe a nonsingular hermitian form of rank 4 over the group ring ℤ[ℤ] which is not extended from the integers. Moreover, we show that under certain indefiniteness asumptions, every nonsingular hermitian form on a free ℤ[ℤ]-module is extended from the integers. As a corollary, there exists a closed oriented 4-dimensional manifold with fundamental group ℤ which is not the connected sum of S¹ × S³ with a simply-connected 4-manifold.