On ruled fields
Journal de Théorie des Nombres de Bordeaux, Volume 1 (1989) no. 1, pp. 27-49.

Some results and problems that arise in connection with the foundations of the theory of ruled and rational field extensions are discussed.

Nous discutons de quelques résultats et problèmes en relation avec les fondements de la théorie des extensions rationnelles de corps d'une ou plusieurs variables.

DOI: 10.5802/jtnb.2
Classification: 13A18,  12F20
Keywords: ruled field extension, uniruled, rational, unirational
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Jack Ohm. On ruled fields. Journal de Théorie des Nombres de Bordeaux, Volume 1 (1989) no. 1, pp. 27-49. doi : 10.5802/jtnb.2. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.2/

[Am, 1955] S.A. Amitsur, Generic splitting fields of central simple algebras, Annals of Math. 62 (1955), 8-43. | MR: 70624 | Zbl: 0066.28604

[A] E. Artin, Algebraic numbers and algebraic functions, Gordon and Breach, New-York (1967). | MR: 237460 | Zbl: 0194.35301

[BCSS, 1985] A. Beauville, J.-L. Colliot-Thelene, J.-J. Sansucand Sir Peter Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles, Ann. of Math. 121 (1986), 283- 318.. | MR: 786350 | Zbl: 0589.14042

[C, 1954] C. Chevalley, On algebraic group varieties, J. Math. Soc. Japan 6 (1954), 305-324. | MR: 67122 | Zbl: 0057.26301

[D,1982] J. Deveney, Ruled function fields, Proc. Amer. Math. Soc. 86 (1982), 213-215. | MR: 667276 | Zbl: 0529.12016

[E] O. Endler, Valuation theory, Springer Verlag, Heidelberg (1972). | MR: 357379 | Zbl: 0257.12111

[K, 1987] M. Kang, The biregular cancellation problem, J. of Pure and Applied Alg. 45 (1987), 241-253. | MR: 890024 | Zbl: 0639.13007

[LT,1952] S. Lang and J. Tate, On Chevalley's proof of Lüroth's theorem, Proc. Amer. Math. Soc. 3 (1952), 621-624. | MR: 49949 | Zbl: 0047.03802

[MH, 1982] T.T. Moh and W. Heinzer, On the Lüroth semigroups and Weierstrass canonical divisors, J. of Alg. 7 (1982), 62-73. | MR: 665164 | Zbl: 0533.14010

[MB,1985] L. Moret-Bailly, Variétés stablement rationnelles non rationnelles, Séminaire Bourbaki (1984 -85). no 643, 14 pp. | Numdam | Zbl: 0605.14044

[N,1967] M. Nagata, A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85-91. | MR: 207688 | Zbl: 0146.26302

[O, 1983] J. Ohm, The ruled residue theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 89 (1983), 16-18. | MR: 706500 | Zbl: 0523.12021

[O, 1984] J. Ohm, On subfields of rational function fields, Arch. Math. 42 (1984), 136-138. | MR: 751484 | Zbl: 0546.12013

[R, 1964] P. Roquette, Isomorphisms of generic splitting fields of simple algebras, J. Reine Angew. Math. 214-215 (1964), 207-226. | MR: 166215 | Zbl: 0219.16023

[Sa, 1953] P. Samuel, Some remarks on Lüroth's theorem, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 27 (1953), 223-224. | MR: 58251 | Zbl: 0053.01902

[Sc] A. Schinzel, Selected topics on polynomials, Univ. of Mich. Press. Ann Arbor, 1982. | MR: 649775 | Zbl: 0487.12002

[Se, 1949] B. Segre, Sur un problème de M. Zariski, Coll. international d'algèbre et de théorie des nombres (Paris, 1949) 135-138. C.N.R.S., Paris, 1950 | MR: 41480 | Zbl: 0040.08201

[Sh] I.R. Shafarevich, Basic algebraic geometry, Grundlehren 213. Springer-Verlag, Heidelberg, 1974. | MR: 366917 | Zbl: 0284.14001

[vdW] B.L. Van Der Waerden, Modern Algebra, Unger. New-York, 1949. | Zbl: 0039.00902

[Wa,1975] W. Waterhouse, The structure of inseparable field extensions, Trans. Amer. Math. Soc. 211 (1975), 39-56. | MR: 379454 | Zbl: 0315.12101

[W] A. Weil, Foundations of algebraic geometry, Coll. Pubs. XXIX, Amer. Math. Soc., Providence (1962). | MR: 144898 | Zbl: 0168.18701

[Z,1958] O. Zariski, On Castelnuovo's criterion of rationality pa = P2 = 0 of an algebraic surface, Illinois J. Math. 2 (1958), 303-315. | MR: 99990 | Zbl: 0085.36203

[ZS 1] O. Zariski and P. Samuel, Commutative algebra. v. 1, van Nostrand, Princeton, 1958. | MR: 90581 | Zbl: 0081.26501

[ZS 2] O. Zariski and P. Samuel, Commutative algebra. v. 2, van Nostrand, Princeton, 1960. | MR: 120249 | Zbl: 0121.27801

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