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Transactions of the American Mathematical Society

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Valuations, primes and irreducibility in polynomial rings and rational function fields

Author: Ron Brown
Journal: Trans. Amer. Math. Soc. 174 (1972), 451-488
MSC: Primary 12J10; Secondary 12A90, 12E05
MathSciNet review: 0371872
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Abstract: The set of extensions of the valuation v on a linearly compact (i.e. maximal) field F to the polynomial ring $ F[x]$ is shown to depend only on the value group and residue class field of v. By a method related to Mac Lane's construction of (rank one) valuations on polynomial rings, a determining invariant is associated with each such extension, called its ``signature". Very roughly, a signature is a pair of sequences, one in the algebraic closure of the residue class field of v and one in the divisible closure of the value group of v. Signatures are also associated with various mathematical objects by means of the extensions of the above sort which naturally arise from them. For example, the set of nonconstant monic irreducible polynomials in $ F[x]$, the set of all finite Harrison primes of the polynomial ring of a global field, and the set of equivalence classes of valuations on the field of rational functions over a global field are each shown to be bijective with a simple set of signatures. Moreover, these objects are studied by means of their associated signatures. For example, necessary and sufficient conditions for irreducibility in $ F[x]$ are given, independent of the language of signatures.

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  • [1] S. Abhyankar, Ramification theoretic methods in algebraic geometry, Ann. of Math. Studies, no. 43, Princeton Univ. Press, Princeton, N. J., 1959. MR 21 #4158. MR 0105416 (21:4158)
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chap. 6: Valuations, Actualités Sci. Indust., no. 1308, Hermann, Paris, 1964. MR 33 #2660. MR 0194450 (33:2660)
  • [3] Ron Brown, Irreducibility over complete rings, Thesis, University of Oregon, Eugene, Ore., 1968.
  • [4] -, Tame extensions of linearly compact fields (in preparation).
  • [5] -, An approximation theorem for extended absolute values, Canad. J. Math. 24 (1972) , 167-184. MR 0292802 (45:1884)
  • [6] Ron Brown and D. K. Harrison, Tamely ramified extensions of linearly compact fields, J. Algebra 15 (1970), 371-375. MR 0262214 (41:6824)
  • [7] P. W. Carruth, Generalized power series fields, Trans. Amer. Math. Soc. 63 (1948), 548-559. MR 9, 561. MR 0024883 (9:561b)
  • [8] J. W. S. Cassels and A. Frölich, Algebraic number theory, Proc. Instructional Conf. Organized by the London Math. Soc. (A NATO Advanced Study Institute), supported by the Internat. Math. Union, Academic Press, London; Thompson, Washington, D. C., 1967. MR 35 #6500. MR 0215665 (35:6500)
  • [9] I. Fleischer, Completeness in valued spaces and algebras, Quart. J. Math. Oxford Ser. (2) 15 (1964), 345-348. MR 31 #2241. MR 0177983 (31:2241)
  • [10] D. K. Harrison, Finite and infinite primes for rings and fields, Mem. Amer. Math. Soc. No. 68 (1966). MR 0207735 (34:7550)
  • [11] H. Inoue, On valuations of polynomial rings of many variables. I, J. Fac. Sci. Hokkaido Univ. Ser. I 21 (1970), 46-74. MR 41 #8410. MR 0263810 (41:8410)
  • [12] I. Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303-321. MR 3, 264. MR 0006161 (3:264d)
  • [13] R. F. MacCoart, Irreducibility of certain classes of Legendre polynomials, Duke Math. J. 28 (1961), 239-246. MR 0123561 (23:A886)
  • [14] R. E. MacKenzie and G. Whaples, Artin-Schreier equations in characteristic zero, Amer. J. Math. 78 (1956), 473-485. MR 19, 834. MR 0090584 (19:834c)
  • [15] S. Mac Lane, A construction for absolute values on polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363-395. MR 1501879
  • [16] -, A construction for prime ideals as absolute values of an algebraic field, Duke Math. J. 2 (1936), 492-510. MR 1545943
  • [17] S. Mac Lane, The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals, Trans. Amer. Math. Soc. 43 (1938), 226-239.
  • [18] -, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122.
  • [19] M. E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. 20 (1969), 193-198. MR 38 #2134. MR 0233813 (38:2134)
  • [20] P. Ribenboim, Théorie des valuations, 2ième éd., Séminaire de Mathématiques Supérieures, no. 9 (Été, 1964), Les Presses de l'Université de Montréal, Montréal, Que., 1968. MR 40 #2670. MR 0249425 (40:2670)
  • [21] O. F. G. Schilling, The theory of valuations, Math. Surveys, no. 4, Amer. Math. Soc., Providence, R. I., 1950. MR 13, 315. MR 0043776 (13:315b)

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Keywords: Valuation, Harrison prime, discrete rank one valuation, polynomial ring, rational function field, irreducibility, Legendre polynomial, Eisenstein criterion, key polynomial, complete field, Henselian field, linearly compact field, maximal field, global field, algebraic extension, ramification, inductive value, limit value
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