Lectures on the Theory of Algebraic Functions of One Variable
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314 Max Deuring Mathematisches Institut der Universit~t G6ttingen, G6ttingen/BRD
Lectures on the Theory of Algebraic Functions of One Variable
Springer-Verlag Berlin-Heidelberg
•
New York 1 9 73
A M S Subject Classifications (1970): 14H05, 1 4 H 10.
I S B N 3-540-06152-5 S p r i n g e r - V e r l a g Berlin • H e i d e l b e r g . N e w Y o r k I S B N 0-387-06152-5 Springer- V e r l a g N e w Y o r k • H e i d e l b e r g • Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1973. Library of Congress Catalog Card Number 72-97679. Printed in Germany, Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE
These
lectures
Research, notes
have been given at the Tata Institute
Bombay,
in ~958.
The supply of the original m i m e o g r a p h e d
having been exhausted
B. Eckmann to republish The aim of these introduction variable,
for some time,
theory of algebraic
to some i n t e r e s t i n g
they still serve this purpose. this new edition all the more
apart
Notes
the proposal
in M a t h e m a t i c s "
aspects
1972
series.
notes by C.P.
of one
of it, and I believe,
It seemed wise not to make
from correcting misprints
since the original
functions
changes
in
and a few minor errors,
Ramanujan
left n o t h i n g
to be desired.
March,
of
was to give a more or less s e l f - c o n t a i n e d
to the algebraic
or, at least,
I accepted
them in the "Lecture
lectures
of F u n d a m e n t a l
M. Deuring,
G~ttingen
TABLE
CHAPTER
I
Lecture
1
Function
-
fields
OF CONTENTS
and v a l u a t i o n s
..................
1
§ I. I n t r o d u c t i o n § 2. O r d e r e d
Lecture
2
§ 3.
3
§ 4. The
Lecture
4
2
groups
§ 3. V a l u a t i o n s , Lecture
places
and v a l u a t i o n
6
valuations
of a r a t i o n a l
of places
§ 6. V a l u a t i o n s
of a l g e b r a i c
§ 7. The
of a place
degree
§ 8. I n d e p e n d e n c e
The R i e m a n n - R o c h
theorem
20
.......................
Lecture
6
§ 10
The
§ 11
The p r i n c i p a l
§ 12
The R i e m a n n
§ 13
Repartitions
§ 14
Differentials
§ 15
The
§ 16
Rational
function
§ 17
Function rational
fields of degree function field
9
20
space
L(¢~)
23
3O 34
Riemann-Roch
theorem
39
fields
43
zero
§ 19. Fields
of genus
one
§ 20. The
CHAPTER
III
-
Zeta
greatest
function
29
theorem
of genus
10
24
divisors
§ 18. Fields Lecture
common
two over
a
11
51 divisor
and L - f u n c t i o n s
§ 22. The
infinite
product
§ 23. The
functional
44 49
of a class
.................
§ 21. The Zeta functi
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