Class numbers of imaginary abelian number fields

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 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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Class numbers of imaginary abelian number fields

Cited by 10 publications

References 33 publications

Fermat’s Last Theorem

Fermat’s Last Theorem

Tame Galois module structure revisited

Class number one problem for normal CM-fields

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