文档介绍：Edited by
W. A. J. LUXEMBURG A. ROBINSON
California Institute of Technology Yale University
1972
NORTH-HOLLAND PANY
AMSTERDAM - LONDON
@ North-Holland pany - 1972
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Library of Congress Catalog Card Number: 76-183275
PRINTED IN HERLANDS
PREFACE
on
on
19 - 25, 1970.
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W.
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B.
VI PREFACE
1971
Abraham ROBINSON
Yale University
1. Introduction. *Q
Q, [
*Q Q. a *Q
Q. a Q.
A = Q(a) c *Q
*Q r
is by F(x, y) = 0 F(x, y) E Q[x, (a,p)
a on *r, a, B
A = Q(a, p)
*Q. F(x, y) = x2 + y2 - 1, r
a2 + p2 = 1 a B
Q(a, B) = Q(o) 0
CI /? by
C? 20 o2- 1
m=- , a=- B=-
1-B o2+ 1, co2+1‘
Q(a, B)
no
r
*Q by
*Q by Q *Q)
by *Q. We
as
in
1) Research supported in part by the National Science Foundation Grant GP-18728-
2 A. ROBINSON
A Q *Q
an internal
valuation of *Q induces a valuation of A and that, moreover, all valuations of
A can be obtained in this way.
In
Q by
Q) K. *K
K. K
by *K by
*K K by
*K K.
2. Valuations induced in A. K *K
K, A K
“K. *K.
*K. x E *K by vpx
x vpx E *Z, *Z
*K.
Suppose now that upx is not identically zero on A - (0). a E A
vptl > 0 (a # j? # 0 A.
vPj?/vPa
a # K
*K vpx = 0 on K - (0)). p
K(tl), f(x, y) E K[x, y]
f(a, p) = 0.
. f(x,v) = 2cijxiy’, cij E K.
p) = 0, cijx$j cklx”y’
. up(cija’fij) = vp(Ckpkp’).
cij ckl K, upcij = vpck, = 0 (i - k)vpa = (I - j)v,p.
I - j # 0 I - j = 0 i - k # 0 vpa = 0,
a.
i-k
. ~pp= ~ Uptl.
1-j
vpp vpa.
. wPx = vpx/uP~
ALGEBRAIC FUNCTION FIELDS 3
A
wpx = 0 x E K - wp
A K. wp
. wp
is induced by