This paper concerns cup product pairings in étale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all positive integers n, there are infinitely many number fields over which there are both trivial and non-trivial Kim invariants associated to cyclic groups of order n.

F.B. was partially supported by NSF FRG Grant DMS-1360621 and NSF Grant DMS-1801328.
T.C. was partially supported by NSF FRG Grants DMS-1265290 and DMS-1360767, NSF SaTC Grants CNS-1513671 and CNS-1701785, Simons Foundation Grant 338379, and NSF Grant DMS-1107263/1107367/1107452 "RNMS: Geometric Structures and Representation Varieties" (the GEAR Network).
R.G. was partially supported by NSF FRG Grant DMS-1360902.
M.K. was partially supported by EPSRC First Grant EP/L021986/1.
G.P. was partially supported by NSF FRG Grants DMS-1360733 and DMS-1701619.