  This paper concerns cup product pairings in \'etale 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 $n > 1$, there are infinitely many number fields $F$ over which there are
both trivial and non-trivial Kim invariants associated to cyclic groups of
order $n$.
