We denote by ℘(x) = x2 +x the characteristic two Artin-Schreier operator. Let
X be the conic defined by the Pfister neighbor φ := [1, b] a of an anisotropic
quadratic Pfister form a, b]]. So the curve X has an affine equation y2 + y + b +
ax2 = 0 and the function field of X is F (X) = F (x, y) where ℘(y) = b + ax2. Let
L be the separable quadratic extension L = F (β) where ℘(β) = b F . Then φ
becomes isotropic over L and the curve XL is rational.
All K-theory groups will be Milnor K-theory. The groups νF (m) and H2
m+1(F
)
are defined by the exact sequence (see [BK])
0 νF (m) ΩF
m

ΩF
m
/dF H2
m+1
(F ) 0.
: ΩF
m
ΩF
m/dF
is defined by ℘(a
df1
f1
···
dfm
fm
) =
(ap
a)
df1
f1
···
dfm
fm
and
νF (m) is ker(℘). Since the characteristic of F is two the Theorems of Kato [K]
asserts that dlog : KmF/2KmF

=
νF (m) is an isomorphism and that H2
m+1(F
)

=
ImWqF/Im+1WqF . The fact that KmF/2KmF

=
νF (m) ΩF m has one im-
mediate and particularly important consequence; if L is a separable extension of
F we necessarily have νF (m) νL(m) is injective as a 2-basis for F remains
2-independent in L.
We use the localization sequences which define the (Milnor) K-cohomology
groups,
0
H0(X,
Km) KmF (X)
p∈X
Km−1F (p)
H1(X,
Km) 0
where the map dF : KmF (X)
p∈X
Km−1F (p) is the sum of tame symbols
corresponding to the valuations vp : F (p) Z at each point p X. Since XL is
rational, results of Milnor [Mi] show that
Hi(XL,
Km)

=
Km−iL for i = 0, 1. When
i = 0 the isomorphism is induced by L L(X) and the second is given by the sum
of norms ΣN : ⊕q∈XL Km−1L(q) Km−1L.
The localization sequences for X and XL fit into the commutative Diagram 1.1
below.
KmF (X)
p∈X
Km−1F (p)

KmL KmL(X)
q∈XL
Km−1L(q) Km−1L

KmF KmF (X)
p∈X
Km−1F (q)
Diagram 1.1.
We remark that in the third column of Diagram 1.1, whenever there is a single
q XL above a point p X, then L(q) is a proper quadratic extension of F (p) and
the first map in the column is induced by inclusion while the second is the norm.
In case there are two q XL above a point p X, then L(q) = F (p) and the maps
are the diagonal and codiagonal. The calculations given in Section 2 take place
within this diagram. The middle row of Diagram 1.1 is exact by Milnor’s result,
but the columns and other row need not be. However, since L/F is quadratic, the
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