## Tesi etd-05232018-222153 |

Thesis type

Tesi di laurea magistrale

Author

FERRI, FABIO

URN

etd-05232018-222153

Title

On Hilbert-Speiser and Leopoldt fields

Struttura

MATEMATICA

Corso di studi

MATEMATICA

Commissione

**relatore**Del Corso, Ilaria

**relatore**Greither, Cornelius

**controrelatore**Dvornicich, Roberto

Parole chiave

- weak normal integral bases
- Hilbert-Speiser
- normal integral bases
- algebraic number theory
- Grothendieck group
- locally free class group
- Galois module structure

Data inizio appello

08/06/2018;

Consultabilità

secretata d'ufficio

Riassunto analitico

In this thesis the object of our study is Galois module structure in the context of number field extensions. It is well known that an extension L/K of number fields has to be tame; the Hilbert-Speiser theorem states that for abelian number fields tameness is equivalent to the existence of a normal integral basis over Q. In a general extension L/K, we know by Noether's theorem that the structure of O_L as an O_K[G]-module makes it locally free with respect to the primes of O_K. Knowing this information, the definition of a locally free class group of modules over O_K[G] leads us to the conclusion that, if G=Gal(L/K) is abelian, L has a NIB over K iff the class of O_L in the class group Cl(O_K[G]) is trivial. This permits us to use these more advanced tools for our problem; for instance, we are going to present the result published by Greither, Replogle, Rubin and Srivastav that Q is the only Hilbert-Speiser number field, i.e. for every bigger number field K there exists a (cyclic of prime order) tame abelian extension that does not have NIB. We will also expose some of the results involving C_l-Hilbert-Speiser fields: a number field K is C_l-Hilbert-Speiser if every cyclic tame extension of order l has NIB. We will mainly focus on necessary conditions for number fields to be C_l-Hilbert-Speiser, exposing some results by Carter, Greither, Herreng, Ichimura, Johnston, Replogle, Yoshimura concerning this.

We present the theory on Hilbert-Speiser fields in the fourth chapter. As we said, we need some theory about locally free class groups. The main framework we need to explore is algebraic K-theory, speaking about the Grothendieck K0(R) and the Whitehead group and K1(R) associated to a ring R. A great part of the earlier results are due to Swan and in the 60’s Milnor noted that from particular commutative diagrams of rings, namely fiber products, we can construct an exact Mayer-Vietoris with Whitehead and Grothendieck groups. From this, under certain hypotheses, Reiner and Ullom concluded that we can restate the Mayer-Vietoris sequence using units and locally free class groups. This is what is very useful for the proof of Greither et al.: the augmentation map ε:O_K[G] → O_K leads to a fiber product whose Mayer-Vietoris sequence will make the study of Cl(O_K[G]) easier. We will expose this theory, in its generality, in the third chapter.

Another important theoretical tool is the study of realizable classes made by McCulloh. He

found a description of the subset of the classes in Cl(O_K[G]) given by the rings of integers O_L when L/K is a tame G-extension, in the case of G elementary abelian (later if G is just abelian). The first idea is to see Cl(O_K[G]) as a Z[∆]-module, where ∆ is the multiplicative group of the finite field whose additive structure is isomorphic to the elementary group G. Inside Z[∆] we find a Stickelberger ideal, which together with the augmentation map allows us to describe the subset of realizable classes, which is actually a subgroup. We will only give some outline of the work by McCulloh in the beginning of the fourth chapter, before proving that this together with Reiner and Ullom’s Mayer-Vietoris implies the main result by Greither et al.

In the last chapter we will study weak normal integral bases: we say that L/K has WNIB if the module O_L is free after having tensored with the maximal order of K[G], which is an O_K-algebra inside K[G] that contains O_K[G]. Here class field theory is very important: in the second chapter we outline it. After having introduced WNIB’s and spoken about an article by Greither which developes some first important techniques, we deal with the problem of finding necessary conditions for a number field K so that every tame cyclic l-extension has WNIB, with l prime number; if this happens we will say that K is C_l-Leopoldt. One may conjecture whether Q continues to be the only field such that every abelian tame extension has WNIB, but there are no proofs so far. The exposed results are not in literature and also give new criteria for C_l-Hilbert-Speiser fields. For example we will see how the study of C_l-Leopoldt fields permits us to correct an oversight contained in an article by

Ichimura, whose overall Hilbert-Speiser-oriented techniques are supple enough to be applied to our problem. In general we obtain restrictions to the class groups and class numbers and finiteness results for number fields K that are linearly disjoint from Q(ζ_l) or intersect it in a certain way. In the very end we see how this is related to Cohen-Lenstra heuristics and Iwasawa main conjecture.

We present the theory on Hilbert-Speiser fields in the fourth chapter. As we said, we need some theory about locally free class groups. The main framework we need to explore is algebraic K-theory, speaking about the Grothendieck K0(R) and the Whitehead group and K1(R) associated to a ring R. A great part of the earlier results are due to Swan and in the 60’s Milnor noted that from particular commutative diagrams of rings, namely fiber products, we can construct an exact Mayer-Vietoris with Whitehead and Grothendieck groups. From this, under certain hypotheses, Reiner and Ullom concluded that we can restate the Mayer-Vietoris sequence using units and locally free class groups. This is what is very useful for the proof of Greither et al.: the augmentation map ε:O_K[G] → O_K leads to a fiber product whose Mayer-Vietoris sequence will make the study of Cl(O_K[G]) easier. We will expose this theory, in its generality, in the third chapter.

Another important theoretical tool is the study of realizable classes made by McCulloh. He

found a description of the subset of the classes in Cl(O_K[G]) given by the rings of integers O_L when L/K is a tame G-extension, in the case of G elementary abelian (later if G is just abelian). The first idea is to see Cl(O_K[G]) as a Z[∆]-module, where ∆ is the multiplicative group of the finite field whose additive structure is isomorphic to the elementary group G. Inside Z[∆] we find a Stickelberger ideal, which together with the augmentation map allows us to describe the subset of realizable classes, which is actually a subgroup. We will only give some outline of the work by McCulloh in the beginning of the fourth chapter, before proving that this together with Reiner and Ullom’s Mayer-Vietoris implies the main result by Greither et al.

In the last chapter we will study weak normal integral bases: we say that L/K has WNIB if the module O_L is free after having tensored with the maximal order of K[G], which is an O_K-algebra inside K[G] that contains O_K[G]. Here class field theory is very important: in the second chapter we outline it. After having introduced WNIB’s and spoken about an article by Greither which developes some first important techniques, we deal with the problem of finding necessary conditions for a number field K so that every tame cyclic l-extension has WNIB, with l prime number; if this happens we will say that K is C_l-Leopoldt. One may conjecture whether Q continues to be the only field such that every abelian tame extension has WNIB, but there are no proofs so far. The exposed results are not in literature and also give new criteria for C_l-Hilbert-Speiser fields. For example we will see how the study of C_l-Leopoldt fields permits us to correct an oversight contained in an article by

Ichimura, whose overall Hilbert-Speiser-oriented techniques are supple enough to be applied to our problem. In general we obtain restrictions to the class groups and class numbers and finiteness results for number fields K that are linearly disjoint from Q(ζ_l) or intersect it in a certain way. In the very end we see how this is related to Cohen-Lenstra heuristics and Iwasawa main conjecture.

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