Let
E
E
be a non-trivial finite Galois extension of a field
K
K
. In this paper we investigate the role that valuation-theoretic properties of
E
/
K
E/K
play in determining the non-triviality of the relative Brauer group,
Br
(
E
/
K
)
\operatorname {Br} (E/K)
, of
E
E
over
K
K
. In particular, we show that when
K
K
is finitely generated of transcendence degree 1 over a
p
p
-adic field
k
k
and
q
q
is a prime dividing
[
E
:
K
]
[E:K]
, then the following conditions are equivalent: (i) the
q
q
-primary component,
Br
(
E
/
K
)
q
\operatorname {Br} (E/K)_{q}
, is non-trivial, (ii)
Br
(
E
/
K
)
q
\operatorname {Br} (E/K)_{q}
is infinite, and (iii) there exists a valuation
π
\pi
of
E
E
trivial on
k
k
such that
q
q
divides the order of the decomposition group of
E
/
K
E/K
at
π
\pi
.