In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper,
definable
will mean definable with parameters.
T
heorem
A.
Let
ℱ = (
F
, +, ×,
V
)
be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset of
F
n
definable in
ℱ
v
.
Then either A is definable in
ℱ = (
F
, +, ×)
or V is definable in
.
T
heorem
B.
Let
ℛ
v
= (
R
, <, +, ×,
V
)
be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset of
R
n
definable in
ℛ
v
.
Then either A is definable in
ℛ = (
R
, <, +, ×)
or V is definable in
.
The proofs of Theorems A and B are quite similar. Both ℱ
v
and ℛ
v
admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(
x, y
) if and only if
v
(
x
) ≤
v
(
y
)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.