Computing all power integral bases in orders of totally real cyclic sextic number fields

Gaál, István

doi:10.1090/s0025-5718-96-00708-9

Cited by 28 publications

(16 citation statements)

References 25 publications

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“…It is known, however, that there are no additional generators of any form for p ^ 13. Specifically, Gaal and Schulte [5] computed all power integral bases for a large collection of cubic fields, including the real cyclotomic field for p = 7; Gaal and Pohst [4] consider a family of totally real cyclic quintic fields and showed that only one, namely the real cyclotomic field for p = 11, admits a power integral basis, and they compute all the generators for this field; Gaal [2] computed all power integral bases for the first five totally real cyclic sextic number fields, which includes the real cyclotomic field for p = 13. In the work of Gaal, Pohst, and Schulte the generators are presented in a different form than in our work, but a close examination shows that we found all the generators for p ^ 13.…”

Section: Now G(x)mentioning

confidence: 99%

Power integral bases for real cyclotomic fields

Miller-Sims¹,

Robertson²

2005

Bull. Austral. Math. Soc.

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A number field K is said to have a power integral basis if its ring of integers OK is of the form Z [a] for some a in the ring. The set of generators is stable under integer translation and multiplication by -1; we call a and a' equivalent if a' = n ± a for some n € Z. Gyory [7] proved that up to equivalence there are only finitely many elements that generate a power basis for any number field K. Most number fields do not have a power integral basis, however, and when one does exist it is usually very difficult to determine all the generators. See Gaal [3] for results on the existence and computation of power integral bases for fields of small degree over Q, and for algorithms for determining power integral bases.Cyclotomic fields are an interesting case because power integral bases always exist and in some cases we can find all the generators (see Nagell [9], Bremner [1], and Robertson [10,11]). Real cyclotomic fields (that is, the maximal real subfields of cyclotomic fields) are also interesting because again power integral bases always exist.Let p ^ 5 be prime and C be a primitive p-th root of unity. Then it is well known that Z[C + C~X] is the ring of integers of the real cyclotomic field Q(C + C~X)-The problem

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Section: Now G(x)mentioning

confidence: 99%

Power integral bases for real cyclotomic fields

Miller-Sims¹,

Robertson²

2005

Bull. Austral. Math. Soc.

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“…In several cases (see, e.g., [9], [10], [11], [12], [13]), this problem was reduced to relative Thue equations.…”

Section: István Gaál and Michael Pohstmentioning

confidence: 99%

On the resolution of relative Thue equations

Gaál

Pohst

2001

Math. Comp.

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Abstract. An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst.Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms.The method is illustrated by numerical examples.

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“…e.g. [3] for the basic ideas of the algorithm) we solved the unit equation (9) corresponding to the index form equation (5). The solutions allow to express ε/ε and hence also ε, which gives (x 2 , x 3 , x 4 , x 5 ) in view of (8), by taking conjugates and solving the corresponding system of linear equations.…”

Section: Lemmamentioning

confidence: 99%

“…e.g. [6], [4], [3], [5]). For number fields of higher degree k the problem becomes difficult because of the large degree k(k − 1)/2 of the index form equation and the number of variables k − 1.…”

Section: Introductionmentioning

confidence: 99%

Power integral bases in a parametric family of totally real cyclic quintics

Gaál

Pohst

1997

Math. Comp.

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Abstract. We consider the totally real cyclic quintic fields Kn = Q(ϑn), generated by a root ϑn of the polynomialAssuming that m = n 4 + 5n 3 + 15n 2 + 25n + 25 is square free, we compute explicitly an integral basis and a set of fundamental units of Kn and prove that Kn has a power integral basis only for n = −1, −2. For n = −1, −2 (both values presenting the same field) all generators of power integral bases are computed.

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Computing all power integral bases in orders of totally real cyclic sextic number fields

Cited by 28 publications

References 25 publications

Power integral bases for real cyclotomic fields

Power integral bases for real cyclotomic fields

On the resolution of relative Thue equations

Power integral bases in a parametric family of totally real cyclic quintics

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