In field theory, the Stufe (/ʃtuːfə/; German: level) s(F) of a field F is the least number of squares that sum to -1. If -1 cannot be written as a sum of squares, s(F)=
∞
. In this case, F is a formally real field. Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.
If
s
(
F
)
≠
∞
then
s
(
F
)
=
2
k
for some
k
∈
N
.
Proof: Let
k
∈
N
be chosen such that
2
k
≤
s
(
F
)
<
2
k
+
1
. Let
n
=
2
k
. Then there are
s
=
s
(
F
)
elements
e
1
,
…
,
e
s
∈
F
∖
{
0
}
such that
0
=
1
+
e
1
2
+
⋯
+
e
n
−
1
2
⏟
=:
a
+
e
n
2
+
⋯
+
e
s
2
⏟
=:
b
.
Both
a
and
b
are sums of
n
squares, and
a
≠
0
, since otherwise
s
(
F
)
<
2
k
, contrary to the assumption on
k
.
According to the theory of Pfister forms, the product
a
b
is itself a sum of
n
squares, that is,
a
b
=
c
1
2
+
⋯
+
c
n
2
for some
c
i
∈
F
. But since
a
+
b
=
0
, we also have
−
a
2
=
a
b
, and hence
−
1
=
a
b
a
2
=
(
c
1
a
)
2
+
⋯
+
(
c
n
a
)
2
,
and thus
s
(
F
)
=
n
=
2
k
.
The Stufe
s
(
F
)
≤
2
for all fields
F
with positive characteristic.
Proof: Let
p
=
char
(
F
)
. It suffices to prove the claim for
F
p
.
If
p
=
2
then
−
1
=
1
=
1
2
, so
s
(
F
)
=
1
.
If
p
>
2
consider the set
S
=
{
x
2
∣
x
∈
F
p
}
of squares.
S
∖
{
0
}
is a subgroup of index
2
in the cyclic group
F
p
×
with
p
−
1
elements. Thus
S
contains exactly
p
+
1
2
elements, and so does
−
1
−
S
. Since
F
p
only has
p
elements in total,
S
and
−
1
−
S
cannot be disjoint, that is, there are
x
,
y
∈
F
p
with
S
∋
x
2
=
−
1
−
y
2
∈
−
1
−
S
and thus
−
1
=
x
2
+
y
2
.
The Stufe s(F) is related to the Pythagoras number p(F) by p(F) ≤ s(F)+1. If F is not formally real then s(F) ≤ p(F) ≤ s(F)+1. The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).
The Stufe of a quadratically closed field is 1.
The Stufe of an algebraic number field is ∞, 1, 2 or 4 ("Siegel's theorem). Examples are Q, Q(√-1), Q(√-2) and Q(√-7).
The Stufe of a finite field GF(q) is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.
The Stufe of a local field of odd residue characteristic is equal to that of its residue field. The Stufe of the 2-adic field Q2 is 4.
