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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on linear automorphisms over $ {\bf R}$

Author: Mowaffaq Hajja
Journal: Proc. Amer. Math. Soc. 111 (1991), 29-34
MSC: Primary 12F20
MathSciNet review: 1036986
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Abstract: Let $ K$ be a rational (= purely transcendental) extension of (the field) $ k$, and let $ s$ be a $ k$-automorphism of $ K$ of finite order. Let $ s$ be linear in the sense that $ K$ has a base $ B$ (i.e., a transcendence basis $ B$ with $ K = k(B)$) for which the $ k$-submodule $ \Sigma (kb:b \in B)$ of $ K$ generated by $ B$ is stabilized by $ s$. In [1, Question 6], it is asked whether $ s$ is completely determined by its order (and $ \operatorname{tr} .{\deg _k}(K)$) and it is proved that, when $ k$ is the complex number field $ {\mathbf{C}}$, then the answer to this question is affirmative iff $ \operatorname{tr} .{\deg _{\mathbf{C}}}(K) > 1$ [1, Corollary 9, Question 6 and Lemma 7]. In this paper, we solve the problem for the field $ {\mathbf{R}}$ of real numbers under the condition that $ \operatorname{tr} .{\deg _{\mathbf{R}}}(K)$ is $ \ne 2,3$. For $ \operatorname{tr} .{\deg _{\mathbf{R}}}(K) = 2\;{\text{or}}\;3$, the problem remains open.

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Article copyright: © Copyright 1991 American Mathematical Society