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Model Theoretic Algebra1

Published online by Cambridge University Press:  12 March 2014

G. L. Cherlin
G. L. Cherlin*
Affiliation:
Universität Heidelberg, Heidelberg, West Germany
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Extract

Since the late 1940's model theory has found numerous applications to algebra. I would like to indicate some of the points of contact between model theoretic methods and strictly algebraic concerns by means of a few concrete examples and typical applications.

§1. The Lefschetz principle. Algebraic geometry has proved to be a fruitful source of model theoretic ideas. What exactly is algebraic geometry? We consider a field K, and let Kn be the set of n-tuples (a 1 … an ) with coordinates ai in K. Kn is called affine n-space over K. Fix polynomials p 1 …, pk in K[x 1, …, xn ] and define

that is V(p 1 …, pk ) is the locus of common zeroes of the pi in Kn . We call V(Pi …, Pk ) the algebraic variety determined by p 1, …, pk . With this terminology we may say:

Algebraic geometry is the study of algebraic varieties defined over an arbitrary field K. This definition lacks both rigor and accuracy, and we will indicate below how it may be improved.

So far we have placed no restrictions on the base field K. Following Weil [4] it is convenient to start with a so-called “universal domain”; in other words take K to be algebraically closed and of infinite transcendence degree over the prime field. Any particular field can of course be embedded in such a universal domain.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 537 - 545
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

Text of a Survey Lecture delivered at the 1974–75 Annual Meeting of the ASL held jointly with the AMS in Washington, D.C., January 1975.

References

REFERENCES

General

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