Let
Q
(
t
)
\mathbb {Q}(t)
be the rational function field over the rationals,
Q
\mathbb {Q}
, let
Q
(
(
t
)
)
\mathbb {Q}((t))
be the Laurent series field over
Q
\mathbb {Q}
, and let
G
\mathcal {G}
be a group of odd order. We investigate the following question: does there exist a finite-dimensional division algebra D central over
Q
(
t
)
\mathbb {Q}(t)
or
Q
(
(
t
)
)
\mathbb {Q}((t))
which is a crossed product for
G
\mathcal {G}
? If such a D exists,
G
\mathcal {G}
is said to be
Q
(
t
)
\mathbb {Q}(t)
-admissible (respectively,
Q
(
(
t
)
)
\mathbb {Q}((t))
-admissible). We prove that if
G
\mathcal {G}
is
Q
(
(
t
)
)
\mathbb {Q}((t))
-admissible, then
G
\mathcal {G}
is also
Q
(
t
)
\mathbb {Q}(t)
-admissible; we also exhibit a
Q
(
t
)
\mathbb {Q}(t)
-admissible group which is not
Q
(
(
t
)
)
\mathbb {Q}((t))
-admissible.