Geodesic
An Approximation Theorem for Extended Prime Spots
Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 167-184

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Brown, Ron. An Approximation Theorem for Extended Prime Spots. Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 167-184. doi: 10.4153/CJM-1972-015-3
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[1] 1. Ron, Brown, Valuations, primes and irreducibility in polynomial rings and rational function fields (to appear). Google Scholar

[2] 2. Ron, Brown and Harrison, D. K., Tamely ramified extensions of linearly compact fields, J. Algebra 15 (1970), 371–375. Google Scholar

[3] 3. Ron, Brown, Extended prime spots and quadratic forms (in preparation). Google Scholar

[4] 4. Fleischer, I., Sur les corps topologiques et les valuations, C. R. Acad. Sci. Paris 286 (1953), 1320–1322. Google Scholar

[5] 5. Griffin, M., Rings of Krull type, J. Reine Angew. Math. 229 (1968), 1–27. Google Scholar

[6] 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] 7. Harrison, D. K. and Warner, H., Infinite primes of fields and completions (to appear). Google Scholar

[8] 8. Hocking, J. and Young, G., Topology (Addison-Wesley, Reading, Massachusetts, 1961). Google Scholar

[9] 9. Kelley, J., General topology (Van Nostrand, Princeton, New Jersey, 1955). Google Scholar

[10] 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] 11. MacLane, S., A construction for absolute values on polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363–395. Google Scholar

[12] 12. Matlis, E., Infective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57–69. Google Scholar

[13] 13. O'Meara, O. T., Introduction to quadratic forms (Academic Press, New York, 1963). Google Scholar

[14] 14. Ribenboim, P., Théorie des valuations (Les Presses de l'Université de Montréal, Montreal, 1964). Google Scholar

[15] 15. Scharlau, W., Quadratic forms, Queen's papers in pure and applied mathematics, No. 22 (Queen's University, Kingston, Ontario, 1969). Google Scholar

[16] 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] 17. Witt, E., Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31–44. Google Scholar

[18] 18. Mayrand-Wong, C., Familles de valuations a charactère compact et théorèmes d'approximation, Ph.D. Dissertation, Université de Montréal, November, 1969. Google Scholar

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