A note on integral bases of unramified cyclic extensions of prime degree. II

Ichimura, Humio; Sumida, Hiroki

doi:10.1007/s002290170039

Cited by 7 publications

(4 citation statements)

References 11 publications

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“…At present, we have numerical examples supporting this conjecture for abelian fields (e.g. [12,14,19,22]). In their methods, cyclotomic units and auxiliary prime numbers are used.…”

Section: Introductionmentioning

confidence: 64%

Computation of the Iwasawa invariants of certain real abelian fields

Sumida-Takahashi

2004

Journal of Number Theory

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Let p be a prime number and k a finite extension of Q: It is conjectured that the Iwasawa invariants l p ðkÞ and m p ðkÞ vanish for all p and totally real number fields k: Some methods to verify the conjecture for each real abelian field k are known, in which cyclotomic units and a set of auxiliary prime numbers are used. We give an effective method, based on the previous one, to compute the exact value of the other Iwasawa invariant n p ðkÞ by using Gauss sums and another set of auxiliary prime numbers. As numerical examples, we compute the Iwasawa invariants associated to k ¼ Qð ffiffi ffi f p ; z p þ z À1 p Þ in the range 1of o200 and 5ppo10 000: r

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“…At present, we have numerical examples supporting this conjecture for abelian fields (e.g. [12,14,19,22]). In their methods, cyclotomic units and auxiliary prime numbers are used.…”

Section: Introductionmentioning

confidence: 64%

Computation of the Iwasawa invariants of certain real abelian fields

Sumida-Takahashi

2004

Journal of Number Theory

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“…Remark 5. It is known by [10][11][12] that the converse of assertion (B) of Childs does not hold in general. This paper is organized as follows.…”

Section: Remarkmentioning

confidence: 99%

“…. u ) = 1 from the first equality in (12). For a with 1 6 a 6 p − 1, we have a ≡ κ i 0 −1 r j mod p for some i 0 and j by (11).…”

Section: Proof Of Theoremmentioning

confidence: 99%

On a Theorem of Childs on Normal Bases of Rings of Integers

Ichimura

2003

J. London Math. Soc.

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Let p be a prime number, F a number field, and H n F the set of all unramified cyclic extensions over F of degree p having a relative normal integral basis. When ζ p ∈ F × , Childs determined the set H n F in terms of Kummer generators. When p = 3 and F is an imaginary quadratic field, Brinkhuis determined this set in a form which is, in a sense, analogous to Childs's result. The paper determines this set for all p > 3 and F with ζ p ∈ F × (and satisfying an additional condition), using the result of Childs and a technique developed by Brinkhuis. Two applications are also given.

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“…Thus, the converse of the assertion (b) does not hold in general. For some related topics on an unramified cyclic extension having a PIB but not a NIB, see [16] and some references therein.…”

Section: Lemma 42 Ifmentioning

confidence: 99%