  We prove a 1966 conjecture of Tate concerning the Artin-Tate pairing on the
Brauer group of a surface over a finite field, which is the analogue of the
Cassels-Tate pairing. Tate asked if this pairing is always alternating and we
find an affirmative answer, which is somewhat surprising in view of the work of
Poonen-Stoll on the Cassels-Tate pairing. Our method is based on studying a
connection between the Artin-Tate pairing and (generalizations of) Steenrod
operations in \'{e}tale cohomology. Inspired by an analogy to the algebraic
topology of manifolds, we develop tools allowing us to calculate the relevant
\'{e}tale Steenrod operations in terms of characteristic classes.
