Model Theoretic Algebra Selected Topics
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521 Greg Cherlin
Model TheoreticAlgebra Selected Topics
Springer-Venag Berlin. Heidelberg- New York lg 76
Author Greg Cherlin Department of Mathematics Rutgers University New Brunswick New Jersey 08903/USA
Ulbrmry of Congress Cataloging in Publication Data
Cherlin, Greg, 1948Model theoretic algebra. (Lecture notes in mathematics ; 521) Bibliography: p. Includes indexes. 1. Model theory. 2. Algebra. I. Title. II~ Series: Lecture notes in mathematics (Berlin) QA3.I28 no. 521 [QAg.7] 510'.8s [512'.02] 76-15388
AMS Subject Classifications (1970): 02 H15 ISBN 3-540-07696-4 Springer-Verlag Berlin Heidelberg 9 - New York ISBN 0-38?-07696-4 Springer-Verlag New York Heidelberg 9 Berlin 9 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 w 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 9Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Table
of C o n t e n t s
Introduction
. . . . . . . . . . . . . . . . . . . .
I
O.
Theory
4
Basic w
I.
Model First
order
w
n-types
w
Unions
w
The
w
Cardinal
w
Definable
Transfer
languages.
and of
First
order
sentences
of
Theorems
The
w
Notes
4
9
9
diagrams
9
theorems
14
. . . . . . . . . . . . . .
9
9
. . . . . . . . . .
in A l g e b r a Maps
Nullstellensatz
~
.
. . . . . . . . . . . . . . . .
and
Hilbert's
seventeenth
21 30
. . . . . . . . . . . . . .
The Ax-~ochen-Ershov over Local Fields)
21 22
problem
. . . . . . . . . . . . . . . .
Exercises
16 17
. . . . . . . . . . . . . .
on C n
9 12
. . . . . . . . . . . . . . . .
sets
w
. o
. . . . . . . . . . . . . . . . . .
transfer
Polynomial
, ~
saturation
Chains
method
w
II.
. . . . . . . . . . . . . . . .
30 (Diophantine
Transfer Principle: . . . . . . . . . . . .
~ 1 7 6
o ~
Problems
o ~
.
.
.
32
.
w
Valued
fields
. . . . . . . . . . . . . .
32
w
p-adic
fields
. . . . . . . . . . . . . .
36
w
Complete
fields
and
cross
Hensel
w
Normalized
w
Artin's
w
Artin-Schreier
theory
w
Puiseux
fields
w
Notes
38
........
44
. . . . . . . . . . . . for
p-adic
o ~
.
.
.
.
.
o
61
Complete
Structures
Existentially
w
Infinitely
w
Finitely
w
Existentially
complete
w
Rings
complete
nilpotents
~6
A generalized
w
Notes
generic generic
without
....
structures
structures
65 ~ 1 7 6
o ,
o ,
. .
74
....
structures
....
commutative
83 92
rings
95
. . . . . . . .
Nullstellensatz
103
......
104
. . . . . . . . . . . . . . . . Complete
Division
division
Rings
105 108
....
109
w
Amalgamating
w w
Existentially
Existentially complete division ring
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