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Splitting fields of conics and sums of squares of rational functions

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Abstract

Given a geometrically unirational variety over an infinite base field, we show that every finite separable extension of the base field that splits the variety is the residue field of a closed point. As an application, we obtain a characterization of function fields of smooth conics in which every sum of squares is a sum of two squares.

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References

  1. Becher K.J., Van Geel J.: Sums of squares in function fields of hyperelliptic curves. Math.Z. 261, 829–844 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Becker, E.: Hereditarily-pythagorean fields and orderings of higher level, Monografias de Mathematica 29, Rio de Janeiro (1978)

  3. Gille, P., Szamuely, T.: Central simple algebras and Galois cohomology. Cambridge studies in advanced mathematics, vol. 101, xi, 343 p. Cambridge University Press, Cambridge (2006)

  4. Hartshorne, R.: Algebraic geometry. Graduate texts in mathematics, vol. 52. Springer, New York (1983)

  5. Lam, T.Y.: Introduction to quadratic forms over fields. Graduate studies in mathematics, vol. 67, AMS, Rhode Island (2004)

  6. Lang, S.: Algebra - revised third edition. Graduate texts in mathematics, vol. 211. Springer, New York, (2002)

  7. Liu, Q.: Algebraic geometry and arithmetic curves. Oxford graduate texts in mathematics, vol. 6. Oxford University Press, Oxford (2002)

  8. Pfister, A.: Quadratic forms with applications to algebraic geometry and topology, vol. 217. Mathematical Society Lecture Note Series, London (1995)

  9. Stichtenoth, H.: Algebraic function fields and codes. 2nd ed. Graduate texts in mathematics, vol. 254. Springer, Berlin (2009)

  10. Tikhonov S.V., Yanchevskiĭ V.I.: Pythagoras numbers of function fields of regular conics over hereditarily pythagorean fields. Dokl. Nats. Akad. Nauk Belarusi 47(2), 5–8 (2003)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Ecole Polytéchnique, Fédérale de Lausanne, MA, C3 1015, Lausanne, Switzerland

    David Grimm

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  1. David Grimm
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Correspondence to David Grimm.

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Grimm, D. Splitting fields of conics and sums of squares of rational functions. manuscripta math. 141, 727–736 (2013). https://doi.org/10.1007/s00229-012-0590-x

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  • DOI: https://doi.org/10.1007/s00229-012-0590-x

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