Equations over Finite Fields An Elementary Approach
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536 Wolfgang M. Schmidt
Equations over Finite Fields An Elementary Approach
Springer-Verlag Berlin-Heidelberg
New 9 York 1976
Author Wolfgang M. Schmidt Department of Mathematics University of Colorado Boulder, Colo., 80309/USA
Library of Congress Cataloging in Publication Data
Sehnddt, Wolfgang M
Equations over finite fields.
(Lecture notes in n~then~tics ; 536) BibliOgraphy: p. i. Diophantine analysis. 2. Modular fields. I. Title. II. Series: Lecture notes in mathematics (Berlin~ ; 536. QAS.Le8 v01.536 [QA242] 510' .8s [512.9'4] 76~6612
A M S Subject Classifications (1970): 10A10,10 B 1 5 , 1 0 G 0 5 , 1 2 C 2 5 , 1 4 G 1 5 ISBN 3 - 5 4 0 - 0 7 8 5 5 - X ISBN 0-387-07855-X
Springer-Verlag Berlin 9Heidelberg 9N e w York Springer-Verlag N e w York 9Heidelberg Berlin 9
This .,,york is subject to copyright. All riclhts 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 s 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. 9 by Springer-Verlag Berlin - Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Preface These a n d K.
Lecture
Notes
Spackman of
I have theorem
tried
on curves
"elementary"
were
lectures to
given
present
over
prepared
a proof
finite
have d i f f e r e n t
at
from notes the as
fields.
University simple
interpretations,
functions
is the one w h i c h o r i g i n a t e d which
as
of
Colorado.
of
of
geometry,
or
that
perhaps
the simplest
Hence
Weil's
"simple"
but I b e l i e v e
in one variable,
with Stepanov.
b y M. R a t l i f f
possible
The notions
for a reader who is u n f a m i l i a r w i t h a l g e b r a i c even w i t h a l g e b r a i c
taken
method
it is this m e t h o d
I follow. T h e l e n g t h of these Notes
be n o t e d that o n l y C h a p t e r s Furthermore,
is p e r h a p s
I and
simplest
case,
simplest
subcase,
when
is c o p r i m e
d
i.e.,
= 0
could
start
to the degree o~
nential
is taken up only there.
sums,
Possible
The
f .
in C h a p t e r
in n u m b e r theory.
This
several
or theorem,
or
results
I w i t h the
first
I do the
special
field and
case is now
The general
on c h a r a c t e r
sums and expo-
stage b e c a u s e os the
Chapters
so
equation
Ill, but a r e a d e r in a hurry
n u m b e r of v a r i a b l e s .
s e q u e n c e s are c h a p t e r s
I, Ill for W e l l ' s
At
to u n d e r g r a d u a t e s .
second chapter,
in an a r b i t r a r y
I by itself,
in C h a p t e r
= f(x)
is i n c l u d e d at such an early
many a p p l i c a t i o n s equations
y
d
and
it should
theorem.
the case w h e n the field is the prime
easy that it could be p r e s e n t e d f(x,y)
leisurely,
I star
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