Class Number Problem for Imaginary Cyclic Number Fields

Chang, Ku-Young; Kwon, Soun-Hi

doi:10.1006/jnth.1998.2298

Cited by 7 publications

(12 citation statements)

References 29 publications

(25 reference statements)

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“…By [18,Theorem 1] if we assume the Generalized Riemann Hypothesis, then d N < (75.08) 48 and hence d N 6 = d 2 K 3 d N 2 < (75.08) 6 and d K 3 < 1.9 × 10 5 .…”

Section: Remark 1 If We Do Not Assume the Generalized Riemann Hypothmentioning

confidence: 98%

“…Let N 2 (N 4 respectively) be the quadratic (quartic respectively) subfield of N 12 . By [6], the conductor f N 2 of N 2 is an odd prime p. 12 has at least three prime divisors and hence g 3. We claim that p i s are ramified in N/N + .…”

Section: Proof Letmentioning

confidence: 98%

“…Suppose that G(N 24 /Q) is abelian. According to [6] and [7] we have G(N 24 By [17] and [38], there is no pair of (N 24.1 , N 24…”

Section: Non-abelian Normal Cm-fields N Of Degree 72 Containing An Ocmentioning

confidence: 99%

“…(All real cubic fields K 3 with d K 3 1.5 × 10 5 and their Hilbert class fields are listed in [46]. If d N 6 < (75.08) 6 …”

Section: Remark 1 If We Do Not Assume the Generalized Riemann Hypothmentioning

confidence: 99%

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Class number one problem for normal CM-fields

Park

Kwon

2007

Journal of Number Theory

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It is known that if we assume the Generalized Riemann Hypothesis, then any normal CM-field with relative class number one is of degree less than or equal to 96. All normal CM-fields of degree less than 48 with class number one are known. In addition, for normal CM-fields of degree 48 the class number one problem is partially solved. In this paper we will show that under the Generalized Riemann Hypothesis there is no more normal CM-fields with class number one except for the possible fields of degrees 64 or 96.

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“…By [18,Theorem 1] if we assume the Generalized Riemann Hypothesis, then d N < (75.08) 48 and hence d N 6 = d 2 K 3 d N 2 < (75.08) 6 and d K 3 < 1.9 × 10 5 .…”

Section: Remark 1 If We Do Not Assume the Generalized Riemann Hypothmentioning

confidence: 98%

Section: Proof Letmentioning

confidence: 98%

“…Suppose that G(N 24 /Q) is abelian. According to [6] and [7] we have G(N 24 By [17] and [38], there is no pair of (N 24.1 , N 24…”

Section: Non-abelian Normal Cm-fields N Of Degree 72 Containing An Ocmentioning

confidence: 99%

“…(All real cubic fields K 3 with d K 3 1.5 × 10 5 and their Hilbert class fields are listed in [46]. If d N 6 < (75.08) 6 …”

Section: Remark 1 If We Do Not Assume the Generalized Riemann Hypothmentioning

confidence: 99%

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Class number one problem for normal CM-fields

Park

Kwon

2007

Journal of Number Theory

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“…These bounds depend on whether L(s, χ) has or does not have a real zero β in the range 0 < β < 1. They will then enable us to obtain in Theorems 18, 28 and 31 bounds for relative class numbers h − K of CM-fields K, especially in the case that K contains an imaginary quadratic subfield L. Apart from the proof of Lemma 17, which can be found in [Lou03], this paper provides the reader with a self-contained exposition of how one can obtain (as in Corollaries 20,21,23 and 25 where several footnotes clearly show that our approach is more efficient than the ones formerly developed by various authors) good enough explicit lower bounds for relative class numbers to enable him to solve various class number problems for CM-fields or to simplify the existing proofs (e.g., see [CK98], [CK00a], [CK00b], [Lou95], [Lou97], [Lou98a], [Lou99], [MM] and [Yam]). Whereas almost all the papers in the literature dealing with explicit lower bounds for relative class numbers of CM-fields (or values at s = 1 of L-functions) use infinite products of Dedekind zeta functions (e.g., see [Bes], [HHRW], [Lou92], [LPP] and [Sta]), our approach is different and stems from integral representations (used to proved functional equations).…”

Section: Introductionmentioning

confidence: 97%

Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields

Louboutin

2006

Acta Arith.

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Tame Galois module structure revisited

Ferri

Greither

2019

Annali di Matematica

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In the last decades a lot of work has been done in the study of Hilbert-Speiser fields; in particular, criteria were developed which insure that a given field is not Hilbert-Speiser. The framework is Galois module structure of extensions of number fields. More precisely the object of our study is the property of an extension of number fields to have a normal integral basis: an extension L/K of number fields admits a NIB if O L is a rank 1 free O K [G]-module. The Hilbert-Speiser theorem states that for abelian number fields tameness is equivalent to the existence of a normal integral basis over Q, while in general we only have that an extension with NIB has to be tame. We will say that a number field K is Hilbert-Speiser if every tame abelian extension L/K has NIB, and C l -Hilbert-Speiser if every such extension with Galois group isomorphic to C l , the cyclic group of prime order l, has NIB.The first contribution to the topic of Hilbert-Speiser fields came from the important result contained in [GRRS99]: Q is the only Hilbert-Speiser field, i.e. for every number field K Q there exists a (cyclic of prime order) tame abelian extension that does not have NIB. Subsequent research went towards the finer problem of finding crieria for C l -Hilbert-Speiser fields. For instance, in [Car03], [Ich04] and [Yos09] there are some conditions for abelian C 2 and C 3 -Hilbert-Speiser fields, while from [Her05] and [GJ09] we know that in general C l -Hilbert-Speiser fields cannot be highly ramified if they are respectively totally imaginary or totally real.In this work, our intention is to consider a weakened version of NIB: we will say that a tame abelian extension L/K of number fields has a weak normal integral basis if M ⊗ OK[G] O L is free of rank 1 over M, where M is the maximal order of K [G]. WNIB's have been studied for instance in [Gre90], [Gre97] and [GJ12]. We shall ask the same questions as above substituting "WNIB" to "NIB" everywhere, and the condition that results from this substitution will be called "Leopoldt field" instead of "Hilbert-Speiser field". We are going to find mainly necessary conditions for number fields to be C l -Leopoldt, where as before l is a prime number, which also give criteria and (sometimes conditional) finiteness results for C l -Hilbert-Speiser fields; for instance we will see that this permits to correct an oversight contained in the article [Ich16], whose techniques, even though they were originally conceived to deal with Hilbert-Speiser fields, turn out to be supple enough to be applied to our problem as well.

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Class Number Problem for Imaginary Cyclic Number Fields

Cited by 7 publications

References 29 publications

Class number one problem for normal CM-fields

Class number one problem for normal CM-fields

Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields

Tame Galois module structure revisited

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