An Elementary Test for the Galois Group of a Quartic Polynomial

Kappe, Luise-Charlotte; Warren, Bette

doi:10.2307/2323198

Cited by 41 publications

(37 citation statements)

References 5 publications

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“…C is the square of an integer. (7.4) This agrees with the deduction of Kappe and Warren [33,Theorem 2].…”

Section: Chaptersupporting

confidence: 91%

“…Moreover, we will establish the simplifying assumptions on the defining quartic of K, g(x) = x 4 + Ax 2 + Bx + C, which are essential in making the proofs contained in this thesis more concise. In Chapter 2 we use the main theorem in [33] discussed above in the case where Gal Q (g(x)) Z/2Z × Z/2Z, which implies that q(x) = x 3 − Ax 2 − 4Cx + 4AC − B 2 splits over Z. Denoting the roots of q(x) as r, s and t, we show that when B 0, the quadratic subfields of K are given by…”

Section: Introductionmentioning

confidence: 99%

“…The approach used in this thesis was made possible by the result stated by Kappe and Warren [33], which gives a powerful and complete classification of Galois groups of the splitting fields defined by irreducible quartic polynomials of the general form g(x) = x 4 + Dx 3 + Ax 2 + Bx + C based entirely on their resolvent cubic q(x) where…”

Section: Introductionmentioning

confidence: 99%

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The Discriminant and Conductor of Bicyclic Quartic Fields

Turner¹

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“…C is the square of an integer. (7.4) This agrees with the deduction of Kappe and Warren [33,Theorem 2].…”

Section: Chaptersupporting

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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The Discriminant and Conductor of Bicyclic Quartic Fields

Turner¹

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“…(1) According to [KW89], if A 2 − B 2 D is not a square, then K is primitive, so this assumption is certainly satisfied if D is prime or squarefree.…”

Section: Remarks 23mentioning

confidence: 99%

Comparing Arithmetic Intersection Formulas for Denominators of Igusa Class Polynomials

Anderson¹,

Balakrishnan²,

Lauter³

et al. 2013

Women in Numbers 2

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Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for CM(K).G 1 under strong assumptions on the ramification in K. Yang later proved this conjecture under slightly stronger assumptions on the ramification. In recent work, Lauter and Viray proved a different formula for CM(K).G 1 for primitive quartic CM fields with a mild assumption, using a method of proof independent from that of Yang. In this paper we show that these two formulas agree, for a class of primitive quartic CM fields which is slightly larger than the intersection of the fields considered by Yang and Lauter and Viray. Furthermore, the proof that these formulas agree does not rely on the results of Yang or Lauter and Viray. As a consequence of our proof, we conclude that the Bruinier-Yang formula holds for a slightly largely class of quartic CM fields K than what was proved by Yang, since it agrees with the Lauter-Viray formula, which is proved in those cases. The factorization of these intersection numbers has applications to cryptography: precise formulas for them allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. The Bruinier-Yang formulaIn this section, we describe the formula for (CM(K).G 1 ) ℓ that was conjectured by Bruinier and Yang [BY06] and later proved by Yang [Yan]. One of the factors (which we denote by R K/ F ) in this intersection formula is multiplicative, so it makes sense to express it in terms of local factors over each prime. The key result of this section is Theorem 2.5, which does 3

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“…From the above observation, we see the following three elementary lemmas (cf. [Buc1910], [Gar28-2], [Les38], [Plo87], [KW89], [JLY02, Chapter 2]):…”

Section: 2mentioning

confidence: 99%