An Algorithm for Computing p-Class Groups of Abelian Number Fields

Aoki, Miho; Fukuda, Takashi

doi:10.1007/11792086_5

Cited by 12 publications

(23 citation statements)

References 10 publications

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“…In another paper by Aoki and Fukuda [4], an algorithm is introduced for the calculation again of the l-part of h + , but for odd primes not dividing the degree of the extension.…”

Section: Introductionmentioning

confidence: 99%

On the class numbers of real cyclotomic fields of conductor pq

Agathocleous

2014

Acta Arith.

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The class numbers h + of the real cyclotomic fields are very hard to compute.Methods based on discriminant bounds become useless as the conductor of the field grows and that is why other methods have been developed, which approach the problem from different angles. In this thesis we extend a method of Schoof that was designed for real cyclotomic fields of prime conductor to real cyclotomic fields of conductor equal to the product of two distinct odd primes. Our method calculates the index of a specific group of cyclotomic units in the full group of units of the field.This index has h + as a factor. We then remove from the index the extra factor that does not come from h + and so we have the order of h + . We apply our method to real cyclotomic fields of conductor < 2000 and we test the divisibility of h + by all primes < 10000. Finally, we calculate the full order of the l-part of h + for all odd primes l < 10000.

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“…In another paper by Aoki and Fukuda [4], an algorithm is introduced for the calculation again of the l-part of h + , but for odd primes not dividing the degree of the extension.…”

Section: Introductionmentioning

confidence: 99%

On the class numbers of real cyclotomic fields of conductor pq

Agathocleous

2014

Acta Arith.

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“…For n = 0, we can obtain full information from Gauss sums of a subfield of K 0 (cf. [1]). However, for n ≥ 1, we cannot directly obtain full information on A ψ * n from Gauss sums (see [7, §4] and the last example of section 3).…”

Section: A Methods Of Computation Of a Nmentioning

confidence: 99%

“…For the last example, it took a few minutes to calculate cyclotomic units modulo prime ideals, and thirty minutes to calculate Gauss sums modulo prime ideals on one PC (CPU: Pentium IV, 3.6GHz, RAM 2GB). In [1], it took 6 hours and 42 minutes to compute A 0 by using Alpha 21264, 667MHz, RAM 4GB.…”

Section: {0}mentioning

confidence: 99%

Computation of the $p$-part of the ideal class group of certain real abelian fields

Sumida-Takahashi

2007

Math. Comp.

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Abstract. Under Greenberg's conjecture, we give an efficient method to compute the p-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the p-part of the ideal class group of the maximal real subfield of Q( √ −f, ζ p n+1 ) in the range 1 < f < 200 and 5 ≤ p < 100000. In order to explain our method, we show an example whose ideal class group is not cyclic.

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“…Neither method from [1] or [9] can deal with this example (p = 2 and p divides the field degree). Both D and f are 40.…”

Section: Application Of P-adic Class Number Formulamentioning

confidence: 99%

“…Whilst there have been approaches to this problem in the past, including attempts by Gras and Gras [8], much progress has been made in the past fifteen years, including most recently work by Hakkarainen [9], which focused on an algorithm to find prime divisors of class numbers, and Aoki and Fukuda [1], whose algorithm was more focused on p-adic decomposition of the class group. Both algorithms require the condition that p does not divide the field degree of K, and p = 2 (problematic as given a fixed degree, genus theory indicates that there are infinitely many fields with class number divisible by the degree).…”

Section: Introductionmentioning

confidence: 99%