A note on linear automorphisms over

Author:
Mowaffaq Hajja

Journal:
Proc. Amer. Math. Soc. **111** (1991), 29-34

MSC:
Primary 12F20

MathSciNet review:
1036986

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Abstract: Let be a rational (= purely transcendental) extension of (the field) , and let be a -automorphism of of finite order. Let be linear in the sense that has a base (i.e., a transcendence basis with ) for which the -submodule of generated by is stabilized by . In [1, Question 6], it is asked whether is completely determined by its order (and ) and it is proved that, when is the complex number field , then the answer to this question is affirmative iff [1, Corollary 9, Question 6 and Lemma 7]. In this paper, we solve the problem for the field of real numbers under the condition that is . For , the problem remains open.

**[1]**Mowaffaq Hajja,*A note on affine automorphisms*, Comm. Algebra**18**(1990), no. 5, 1535–1549. MR**1059746**, 10.1080/00927879008823981**[2]**M. Hajja,*The alternating functions of three and of four variables*, Algebras Groups Geom.**6**(1989), no. 1, 49–54. MR**1023463****[3]**Mowaffaq Hajja and Reyadh Khazal,*A note on algebraic automorphisms*, J. Algebra**139**(1991), no. 2, 336–341. MR**1113779**, 10.1016/0021-8693(91)90297-L**[4]**Nathan Jacobson,*Basic algebra. II*, 2nd ed., W. H. Freeman and Company, New York, 1989. MR**1009787****[5]**H. W. Lenstra Jr.,*Rational functions invariant under a cyclic group*, Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979) Queen’s Papers in Pure and Appl. Math., vol. 54, Queen’s Univ., Kingston, Ont., 1980, pp. 91–99. MR**634683**

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1036986-9

Article copyright:
© Copyright 1991
American Mathematical Society