On totally real Hilbert–Speiser fields of type C<sub>p</sub>

Greither, Cornélius; Johnston, Henri

doi:10.4064/aa138-4-3

Cited by 4 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We can often verify part (b) of Proposition 1.8 by using the results of [1] and [2]. In other cases, we can quote existing results in the literature showing that K is not Hilbert-Speiser of type C p (see [12], [14]). Combining all of these results gives the following theorem.…”

Section: We Observe Immediately That Trmentioning

confidence: 84%

See 1 more Smart Citation

On the restricted Hilbert–Speiser and Leopoldt properties

Byott¹,

Carter²,

Greither³

et al. 2011

Illinois J. Math.

Self Cite

View full text Add to dashboard Cite

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O L is free as an O K [G]-module. If O L is free over the associated order A L/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G = C p has prime order. We give examples with G = C p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

show abstract

Section: We Observe Immediately That Trmentioning

confidence: 84%

“…The above-mentioned work of Miyata allows us to prove the following key proposition, which in particular shows that global failure of mHS-L(K, C p ) cannot occur. With this in mind, we point out the main result of [12]. Its proof is based on a detailed analysis of locally free class groups and ramification indices.…”

Section: We Observe Immediately That Trmentioning

confidence: 97%

On the restricted Hilbert–Speiser and Leopoldt properties

Byott¹,

Carter²,

Greither³

et al. 2011

Illinois J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…If l were not ramified in K, the extension K(ζ l )/K( √ −l) would satisfy the hypotheses of [Was97, Theorem 10.1], and so the norm map between class groups would be surjective. This is what Ichimura considered (indeed we already know that totally real C l -Hilbert-Speiser fields are not ramified in l, by [GJ09]), but actually for our purposes we can just assume nonarithmetic disjointness: by the well-known commutative the diagram given by Artin maps, namely…”

Section: Totally Real C L -Leopoldt Fieldsmentioning

confidence: 90%

“…When instead [Q(ζ l ) : K ∩ Q(ζ l )] is even, i.e. K ∩ Q(ζ l ) is real, then if K is C l -Hilbert-Speiser the intersection is trivial for l ≥ 7, by [GJ09].…”

Section: Complete List Of Real Abelianmentioning

confidence: 99%

Tame Galois module structure revisited

Ferri

Greither

2019

Annali di Matematica

Self Cite

View full text Add to dashboard Cite

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.

show abstract