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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rationality of some genus 0 extensions of $ K(X)$


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 $ L$ be a quadratic extension of a field $ K$ with Galois group $ \left\{ {e,\Gamma } \right\}$. Let $ \left\{ {x,y} \right\}$ be algebraically independent over $ L$ and let $ \Gamma $ be extended to an automorphism of $ L(x,y)$ so that $ \Gamma (x) = x$ and the extension has order 2. Then $ L{(x,y)^\Gamma }$ is a genus 0 extension of $ K(x)$. This paper examines when $ L{(x,y)^\Gamma }$ will be pure transcendental over $ K$. It is shown that some important examples from field theory can be realized by this construction. The main result shows that $ L{(x,y)^\Gamma }$ is pure transcendental over $ K$ if $ \Gamma (y) = ({x^2} + bx + c)/y(\operatorname{char} K \ne 2)$. An example illustrates that it is essential that the second degree polynomial be monic.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1007494-5
PII: S 0002-9939(1990)1007494-5
Article copyright: © Copyright 1990 American Mathematical Society