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Computer Mathematics pp 67–85Cite as

On the Simplest Quartic Fields and Related Thue Equations

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Abstract

Let \(K\) be a field of char \(K\ne 2\). For \(a\in K\), we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial \(X^4-\,aX^3-\,6X^2+\,aX+\,1\) over \(K\) as the special case of the field intersection problem via multi-resolvent polynomials. From this result, over an infinite field \(K\), we see that the polynomial gives the same splitting field over \(K\) for infinitely many values \(a\) of \(K\). We also see by Siegel’s theorem for curves of genus zero that only finitely many algebraic integers \(a\in \fancyscript{O}_K\) in a number field \(K\) may give the same splitting field. By applying the result over the field \(\mathbb {Q}\) of rational numbers, we establish a correspondence between primitive solutions to the parametric family of quartic Thue equations

$$ X^4-mX^3Y-6X^2Y^2+mXY^3+Y^4=c, $$

where \(m\in \mathbb {Z}\) is a rational integer and \(c\) is a divisor of \(4(m^2+16)\), and isomorphism classes of the simplest quartic fields.

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Acknowledgments

This work was partially supported by Rikkyo University Special Fund for Research.

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Authors and Affiliations

  1. Department of Mathematics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

    Akinari Hoshi

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  1. Akinari Hoshi
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Correspondence to Akinari Hoshi .

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Editors and Affiliations

  1. Academy of Mathematics & System Science, Beijing, China

    Ruyong Feng

  2. Department of Mathematics and Computer S, University of Antwerp, Antwerp, Antwerpen, Belgium

    Wen-shin Lee

  3. Department of Mathematical Information S, Tokyo University of Science, Tokyo, Tokyo, Japan

    Yosuke Sato

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Hoshi, A. (2014). On the Simplest Quartic Fields and Related Thue Equations. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_7

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