On the Simplest Quartic Fields and Related Thue Equations

Akinari Hoshi ()
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Akinari Hoshi: Rikkyo University, Department of Mathematics

A chapter in Computer Mathematics, 2014, pp 67-85 from Springer

Abstract: Abstract Let $$K$$ K be a field of char $$K\ne 2$$ K ≠ 2 . For $$a\in K$$ a ∈ K , we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial $$X^4-\,aX^3-\,6X^2+\,aX+\,1$$ X 4 - a X 3 - 6 X 2 + a X + 1 over $$K$$ K as the special case of the field intersection problem via multi-resolvent polynomials. From this result, over an infinite field $$K$$ K , we see that the polynomial gives the same splitting field over $$K$$ K for infinitely many values $$a$$ a of $$K$$ K . We also see by Siegel’s theorem for curves of genus zero that only finitely many algebraic integers $$a\in \fancyscript{O}_K$$ a ∈ O K in a number field $$K$$ K may give the same splitting field. By applying the result over the field $$\mathbb {Q}$$ Q of rational numbers, we establish a correspondence between primitive solutions to the parametric family of quartic Thue equations $$ X^4-mX^3Y-6X^2Y^2+mXY^3+Y^4=c, $$ X 4 - m X 3 Y - 6 X 2 Y 2 + m X Y 3 + Y 4 = c , where $$m\in \mathbb {Z}$$ m ∈ Z is a rational integer and $$c$$ c is a divisor of $$4(m^2+16)$$ 4 ( m 2 + 16 ) , and isomorphism classes of the simplest quartic fields.

Keywords: Thue Equations; Primitive Solution; Field Splitting; Explicit Answer; Resolvent Polynomial (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-43799-5_7

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DOI: 10.1007/978-3-662-43799-5_7

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