Abstract
One of the first and most fundamental results in the theory of non-algebraic field extensions is Lüroth’s theorem (1876): If k ⊂ L ⊆ k(t) = K are field extensions, with t transcendental over k, then there exists u ε L such that L = k(u).
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References
S. S. Abhyankar and C. Bajaj, Computations with algebraic curves; Symbolic and algebraic computation, ISSAC ’88; Lecture notes in computer science, No. 358, Springer, 1989, pp. 274–284.
S. A. Amitsur, Generic splitting fields of central simple algebras, Annals of Math. 62 (1955), 8–43.
S. A. Amitsur, Generic splitting fields; Brauer groups in ring theory and algebraic geometry (F. van Oystaeyen and A. Verschoren editors), Lecture Notes in Math. 917, Springer, Berlin, 1982, pp. 1–24.
S. A. Amitsur, Division algebras — a survey, Contemporary Math. 13 (1982), 3–26.
E. Artin, Algebraic numbers and algebraic functions, Gordon and Breach, New York, 1967.
A. Beauville, J.-L. Colliot-Théléne, J.-J. Sansuc, and Sir P. Swinnerton-Dyer, Varietes stablement rationnelles non rationnelles, Annals of Math. 121 (1986), 283–315.
C. Chevalley, Introduction to the theory of algebraic functions of one variable, Math. Surveys 6, American Math. Soc., New York, 1951.
J. -L. Colliot-Thélène, Les grands themes de Francois Chatelet, L’Enseignement Math. 34 (1988), 387–405.
J. -L. Colliot-Thélène, Arithmetic des variétés rationnelles et problèmes birationnels, Proc. Int. Congress Math. Berkeley, 1986, 641–653.
J. -L. Colliot-Thélène, J.-J. Sansuc, and Sir P. Swinnerton-Dyer, Intersections of two quadrics and Chatelet surfaces, J. fur die reine u. angew. Math.; Part I, vol. 373 (1986), 37–107; Part II, vol. 374 (1986), 72–165.
D. Coray, The Hasse principle for pairs of quadratic forms, in: Journees arithmétiques 1980, ed. J. V. Armitage, London Math. Soc. Lecture Notes Series 56 (1982), 237–246.
J. Deveney, Ruled function fields, Proc. Amer. Math. Soc. 86 (1982), 213–215.
M. Eichler, Introduction to the theory of algebraic numbers and functions, Academic Press, New York, 1966.
M. Fried, A note on principal ideals and smooth curves, J. of Algebra 74 (1982), 124–139.
W.-D. Geyer, The automorphism group of the field of all algebraic numbers, Atas da 5a escola de algebra, IMPA vol. 11, Rio de Janeiro, 1978, 167–199.
I. Kaplansky, Linear algebra and geometry, Chelsea, New York, 1974.
I. Kersten, Brauergruppen von Korpern, Aspekte der Math. Band D6, Vieweg & Sohn, Braunschweig/Wiesbaden, 1990.
M. Knebusch, Generic splitting of quadratic forms I, Proc. London Math. Soc. 33 (1976), 65–93; II, ibid. 34 (1977), 1–31.
T. Y. Lam, The algebraic theory of quadratic forms, Benjamin, Reading, Mass., 1980.
S. Lang and J. Tate, On Chevalley’s proof of Lüroth’s theorem, Proc. Amer. Math. Soc. 3 (1952), 621–624.
R. E. MacRae, On rational points on conics, Proc. Amer. Math. Soc. 67 (1977), 38–40.
R. E. MacRae and P. Samuel, Subfields of index 2 of elliptic function fields, Conference on commutative algebra, Lawrence, Kansas 1972, 171–193; Lecture notes in math. 311, Springer, Berlin, 1973.
Yu. I. Manin and M. A. Tsfasman, Rational varieties: algebra, geometry and arithmetic, Russian Math. Surveys 41:2 (1986), 51–114.
T. T. Moh and W. Heinzer, On the Lüroth semigroup and Weier-strass canonical divisors, J. of Algebra 77 (1982), 62–73.
L. Moret-Bailly, Variétés stablement rationnelles non rationnelles, Sem. Bourbaki, 1984–85, no. 643, 14 pp.
M. Nagata, A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85–91.
M. Nagata, Field theory, Dekker, New York, 1977.
J. Ohm, On ruled fields, Sém. de théorie des nombres, Univ. Bordeaux 1 (1988–1989), 27–49.
L. Roberts, K1 of a curve of genus zero, Trans. Amer. Math. Soc. 188 (1974), 319–324.
P. Samuel, Some remarks on Lüroth’s theorem, Memoirs of College of Sci. of Univ. of Kyoto 27 (1953), 223–224.
P. Samuel, Lectures on old and new results on algebraic curves, Tata Institute, Bombay, 1964.
P. Samuel, Algebraic theory of numbers, Hermann, Paris, 1970.
B. Segre, Sur un problem de M. Zariski, Coll. international d’algebre et de theorie des nombres (Paris, 1949) 135–138, C. N. R. S., Paris, 1950.
J. -P. Serre, A course in arithmetic, Springer, New York, 1973.
J.-P. Serre, Local fields, Springer, New York, 1979.
A. A. Suslin, The quaternion homomorphism for the function field on a conic, Soviet Math. Doklady 26 (1982), 72–77.
R. Swan, Noether’s problem in galois theory; Emmy Noether in Bryn Mawr (B. Srinivasan and J. Sally, editors), Springer, 1983, pp. 21–40.
J. Tate, Genus change in inseparable extensions of function fields, Proc. Amer. Math. Soc. 3 (1952), 400–404.
B. L. van der Waerden, Modem algebra I, Unger, New York, 1949.
A. Wadsworth, Merkurjev’s elementary proof of Merkurjev’s theorem, Contempory Math. Vol. 55, Part II (1986), 741–774.
A. Wadsworth, Letter, May 6, 1990.
A. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields, Trans. Amer. Math. Soc. 208 (1975), 352–358
W. Waterhouse, The structure of inseparable field extensions, Trans. Function Fields of Conies, a Theorem, & a Problem 363 Amer. Math. Soc. 211 (1975), 39–54.
A. Weil, Foundations of algebraic geometry, Amer. Math. Soc. Colloq. Pub. 29, Providence, 1962.
O. Zariski and P. Samuel, Commutative algebra, vol. I, van Nostrand, Princeton, 1955.
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Ohm, J. (1994). Function Fields of Conies, a Theorem of Amitsur—MacRae, and a Problem of Zariski. In: Bajaj, C.L. (eds) Algebraic Geometry and its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2628-4_21
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DOI: https://doi.org/10.1007/978-1-4612-2628-4_21
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