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An Approximation Theorem for Extended Prime Spots

Published online by Cambridge University Press:  20 November 2018

Ron Brown
Ron Brown*
Affiliation:
University of Oregon, Eugene, Oregon; Simon Fraser University, Burnaby, British Columbia
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We introduce here a generalization to arbitrary fields of the prime spots of algebraic number theory, essentially by allowing absolute values to take the value ∞. The set of “extended” prime spots of a field admits a natural topology, and an approximation theorem is given here for compact sets of extended prime spots. Among the corollaries of the approximation theorem are the weak approximation theorem for absolute values [13, p. 8], Ribenboim's generalization of the approximation theorem for independent valuations [14, p. 136], Stone's approximation theorem for type V topologies [16, p. 20], and an approximation theorem for Harrison primes.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 167 - 184
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Ron, Brown, Valuations, primes and irreducibility in polynomial rings and rational function fields (to appear).Google Scholar
2. Ron, Brown and Harrison, D. K., Tamely ramified extensions of linearly compact fields, J. Algebra 15 (1970), 371–375.Google Scholar
3. Ron, Brown, Extended prime spots and quadratic forms (in preparation).Google Scholar
4. Fleischer, I., Sur les corps topologiques et les valuations, C. R. Acad. Sci. Paris 286 (1953), 1320–1322.Google Scholar
5. Griffin, M., Rings of Krull type, J. Reine Angew. Math. 229 (1968), 1–27.Google Scholar
6. Harrison, D. K., Finite and infinite primes for rings and fields, Memoirs Amer. Math. Soc. No. 68 (Amer. Math. Soc, Providence, R.I., 1966).Google Scholar
7. Harrison, D. K. and Warner, H., Infinite primes of fields and completions (to appear).Google Scholar
8. Hocking, J. and Young, G., Topology (Addison-Wesley, Reading, Massachusetts, 1961).Google Scholar
9. Kelley, J., General topology (Van Nostrand, Princeton, New Jersey, 1955).Google Scholar
10. Knebusch, M., Rosenberg, A., and Ware, R., Structure of Witt rings, quotients of abelian group rings, and orderings of fields, Bull. Amer. Math. Soc. 77 (1971), 205–210.Google Scholar
11. MacLane, S., A construction for absolute values on polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363–395.Google Scholar
12. Matlis, E., Infective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57–69.Google Scholar
13. O'Meara, O. T., Introduction to quadratic forms (Academic Press, New York, 1963).Google Scholar
14. Ribenboim, P., Théorie des valuations (Les Presses de l'Université de Montréal, Montreal, 1964).Google Scholar
15. Scharlau, W., Quadratic forms, Queen's papers in pure and applied mathematics, No. 22 (Queen's University, Kingston, Ontario, 1969).Google Scholar
16. Stone, A. L., Nonstandard analysis in topological algebra, from Applications of Model Theory to Algebra, Analysis and Probability, edited by Luxemburg, W. (Holt, Rinehart and Winston, New York, 1969).Google Scholar
17. Witt, E., Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31–44.Google Scholar
18. Mayrand-Wong, C., Familles de valuations a charactère compact et théorèmes d'approximation, Ph.D. Dissertation, Université de Montréal, November, 1969.Google Scholar