Any finite degree field extension
K
/
F
K/F
determines an ideal
T
K
/
F
{\mathcal {T}_{K/F}}
of the Witt ring
W
F
WF
of
F
F
, called the transfer ideal, which is the image of any nonzero transfer map
W
K
→
W
F
WK \to WF
. The ideal
T
K
/
F
{\mathcal {T}_{K/F}}
is computed for certain field extensions, concentrating on the case where
K
K
has the form
F
(
a
1
,
…
,
a
n
)
F\left ({\sqrt {{a_1}} , \ldots ,\sqrt {{a_n}} } \right )
,
a
i
∈
F
{a_i} \in F
. When
F
F
and
K
K
are global fields, we investigate whether there is a local global principle for membership in
T
K
/
F
{\mathcal {T}_{K/F}}
. This is shown to be equivalent to the existence of a "Hasse norm theorem mod squares," i.e., a local global principle for the image of the norm map
N
K
/
F
:
K
∗
/
K
∗
2
→
F
∗
/
F
∗
2
{N_{K/F}}: {K^\ast }/{K^{\ast 2}} \to {F^\ast }/{F^{\ast 2}}
. It is shown that such a Hasse norm theorem holds whenever
K
=
F
(
a
1
,
…
,
a
n
)
K = F(\sqrt {a_1},\ldots ,\sqrt {a_n})
, although it does not always hold for more general extensions of global fields, even some Galois extensions with group
Z
/
2
Z
×
Z
/
4
Z
\mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/4\mathbb {Z}
.