Rationality of some genus 0 extensions of

Authors:
James K. Deveney and Joe Yanik

Journal:
Proc. Amer. Math. Soc. **109** (1990), 53-58

MSC:
Primary 12F20

MathSciNet review:
1007494

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Abstract: Let be a quadratic extension of a field with Galois group . Let be algebraically independent over and let be extended to an automorphism of so that and the extension has order 2. Then is a genus 0 extension of . This paper examines when will be pure transcendental over . It is shown that some important examples from field theory can be realized by this construction. The main result shows that is pure transcendental over if . An example illustrates that it is essential that the second degree polynomial be monic.

**[1]**Arnaud Beauville, Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Peter Swinnerton-Dyer,*Variétés stablement rationnelles non rationnelles*, Ann. of Math. (2)**121**(1985), no. 2, 283–318 (French). MR**786350**, 10.2307/1971174**[2]**James K. Deveney and Joe Yanik,*Nonrational fixed fields*, Pacific J. Math.**139**(1989), no. 1, 45–51. MR**1010783****[3]**E. Formanek,*Rational function fields. Noether’s problem and related questions*, J. Pure Appl. Algebra**31**(1984), no. 1-3, 28–36. MR**738202**, 10.1016/0022-4049(84)90073-2**[4]**Mowaffaq Hajja,*Rationality of finite groups of monomial automorphisms of 𝑘(𝑥,𝑦)*, J. Algebra**109**(1987), no. 1, 46–51. MR**898335**, 10.1016/0021-8693(87)90162-1**[5]**Peter Roquette,*On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras*, Math. Ann.**150**(1963), 411–439. MR**0154888****[6]**David J. Saltman,*Generic Galois extensions and problems in field theory*, Adv. in Math.**43**(1982), no. 3, 250–283. MR**648801**, 10.1016/0001-8708(82)90036-6**[7]**Deborah Diller Triantaphyllou,*Invariants of finite groups acting nonlinearly on rational function fields*, J. Pure Appl. Algebra**18**(1980), no. 3, 315–331. MR**593621**, 10.1016/0022-4049(80)90007-9

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1007494-5

Article copyright:
© Copyright 1990
American Mathematical Society