A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

Hoshi, Akinari; Miyake, Katsuya

doi:10.1007/978-93-86279-46-0_7

Cited by 7 publications

(19 citation statements)

References 45 publications

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“…In Section 3, by using multi-resolvent polynomials, we will give an explicit form of an answer to the field isomorphism problem of f s (X) over K as the special case of the field intersection problem (cf. the simplest cubic case [Mor94], [Cha96], [Oga03], [Kom04], [HM09a], [H]). One of the advantages of using multi-resolvent polynomials is the validity for non-abelian groups (see [HM07], [HM09b], [HM09c], [HM]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On the Simplest Quartic Fields and Related Thue Equations

Hoshi

2014

Computer Mathematics

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Let K be a field of char K = 2. For 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 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 ∈ OK in a number field K may give the same splitting field. By applying the result over the field Q of rational numbers, we establish a correspondence between primitive solutions to the parametric family of quartic Thue equationswhere m ∈ 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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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…However the multi-resolvent polynomial R 1 f a,b (X) = RP Θ1,U,f a,b (X) where f a,b (X) = f a (X)f b (X) becomes complicated in the both cases above. (i) In [HM09b], we gave an answer to the field isomorphism problem of f s (X) by taking…”

Section: Preliminariesmentioning

confidence: 99%

On the Simplest Quartic Fields and Related Thue Equations

Hoshi

2014

Computer Mathematics

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“…Theorem 4.3 is a generalization of the results of the simplest cubic (resp. quartic) case in [Mor94], [Cha96], [HM09a] (resp. [H2]).…”

Section: By Theorem 34 We Get the Intersection Field Splmentioning

confidence: 99%

On the simplest sextic fields and related Thue equations

Hoshi¹

2012

Funct. Approx. Comment. Math.

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“…In the cyclic cubic case, Morton [10] gave an explicit answer to this problem for the generic polynomial X 3 + sX 2 − (s + 3)X + 1 for C 3 over a field K with char K = 2 (see also [1] and [3]). In [2], the authors investigated the field isomorphism problem of f s (X) over a field K with char K = 3 and gave the following theorem: Corollary 8]).…”

Section: Introductionmentioning

confidence: 99%