Power Bases for 2-Power Cyclotomic Fields

Robertson, Leanne

doi:10.1006/jnth.2000.2621

Cited by 11 publications

(8 citation statements)

References 18 publications

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“…The proof follows the proof in the 2-power case given in Robertson [10]. Let σ g repeatedly act on the equation σ g (μ) = μλ.…”

Section: Lemma 21 Let λ and μ Be Units Inmentioning

confidence: 94%

“…In Robertson [10] we considered the case where ζ is a primitive 2 m th root of unity and showed that if λ is a cyclotomic unit then μ is a cyclotomic unit as well. In the current case ζ is a primitive p m th root of unity and we require the additional condition that gcd(h + q , p(p − 1)/2) = 1.…”

Section: Cyclotomic Unitsmentioning

confidence: 99%

“…In the case of 2-power cyclotomic fields, there are no generators on the line Re(z) = 1/2 and all the generators are equivalent to ζ (see [10]). In the current case of p-power cyclotomic fields where p is an odd prime, there do exist generators for O q on the line Re(z) = 1/2.…”

Section: Introductionmentioning

confidence: 98%

“…See Bremner [1] and Robertson [9] for a study of power integral bases in prime cyclotomic fields. See Robertson [10] for the determination of all power integral bases in 2-power cyclotomic fields. In this paper we study power integral bases in p-power cyclotomic fields for odd primes p.…”

Section: Introductionmentioning

confidence: 99%

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Power integral bases in prime-power cyclotomic fields

Gaál

Robertson

2006

Journal of Number Theory

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Let p be an odd prime and q = p m , where m is a positive integer. Let ζ be a primitive qth root of unity, and O q be the ring of integers in the cyclotomic field Q(ζ ). We prove that if O q = Z[α] and gcd(h + q , p(p − 1)/2) = 1, where h + q is the class number of Q(ζ + ζ −1 ), then an integer translate of α lies on the unit circle or the line Re(z) = 1/2 in the complex plane. Both are possible since O q = Z[α] if α = ζ or α = 1/(1 + ζ ). We conjecture that, up to integer translation, these two elements and their Galois conjugates are the only generators for O q , and prove that this is indeed the case when q = 25.

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“…The proof follows the proof in the 2-power case given in Robertson [10]. Let σ g repeatedly act on the equation σ g (μ) = μλ.…”

Section: Lemma 21 Let λ and μ Be Units Inmentioning

confidence: 94%

Section: Cyclotomic Unitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Power integral bases in prime-power cyclotomic fields

Gaál

Robertson

2006

Journal of Number Theory

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“…Alors soit α est équivalent à ζ q , soit α est équivalent à 1/(ζ q + 1). Robertson [8] démontre le théorème suivant qui preuve une telle conjecture dans le cas où q est une puissance de 2 : Ensuite, Robertson et Gaál [3] obtiennent un résultat similaire au théorème 1 pour les classes des générateurs de Z[ζ q ], où q est la puissance d'un nombre premier p, plus précisément :…”

Section: Introductionunclassified

Générateurs de l'anneau des entiers d'une extension cyclotomique

Ranieri

2008

Journal of Number Theory

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Soit p un nombre premier impair, soit q = p m , où m est un entier positif ; notons ζ q une racine primitive q-ième de l'unité et O q l'anneau des entiers de Q(ζ q ). Dans [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372-384] I. Gaál et L. Robertson montrent que si (h + q , p(p − 1)/2) = 1, où h + q est l'ordre du groupe des classes de Q(ζ q + ζ q ), alors si α ∈ O q engendre O q (autrement dit Z[α] = O q ) soit α est un conjugué d'un translaté par un entier de ζ q soit α + α est un entier impair. Dans notre travail nous montrons que nous pouvons éliminer l'hypothèse sur h + q . Autrement dit nous prouvons que si α ∈ O q engendre O q soit α est un conjugué d'un translaté par un entier de ζ q soit α + α est un entier impair. AbstractLet p be an odd prime and q = p m , where m is a positive integer. Let ζ q be a qth primitive root of 1 and O q be the ring of integers in Q(ζ q ). In [I. Gaál, L. Robertson, Power integral bases in primepower cyclotomic fields, J. Number Theory 120 (2006) 372-384] I. Gaál and L. Robertson show thateither α is equals to a conjugate of an integer translate of ζ q or α + α is an odd integer. In this paper we show that we can remove the hypothesis over h + q . In other words we show that if α ∈ O q is a generator of O q then either α is a conjugate of an integer translate of ζ q or α + α is an odd integer.

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Some Higher Degree Fields

Gaál

2002

Diophantine Equations and Power Integral Bases

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Power Bases for 2-Power Cyclotomic Fields

Cited by 11 publications

References 18 publications

Power integral bases in prime-power cyclotomic fields

Power integral bases in prime-power cyclotomic fields

Générateurs de l'anneau des entiers d'une extension cyclotomique

Some Higher Degree Fields

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