On Hilbert-Speiser and Leopoldt fields

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Universit`

a degli Studi di Pisa

FACOLT `A DI MATEMATICA Corso di Laurea in Matematica

Tesi di laurea magistrale

On Hilbert-Speiser and Leopoldt fields

Candidato Fabio Ferri

Relatori

Prof.ssa Ilaria Del Corso Prof. Cornelius Greither Controrelatore

Prof. Roberto Dvornicich

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Introduction

In this thesis the object of our study is Galois module structure in the context of number field extensions. A classical question that arose in the last century is whether in a Galois extension L/K of number fields with Galois group G = Gal(L/K) there exists an element

α ∈ OL such that its conjugates form a free OK-basis of OL, i.e. OL is a rank 1 free OK[G]-module; in this case we say that the extension has a normal integral basis (NIB for short). This arithmetic property came up from the Galois-theoretic well known fact that in any finite Galois extension of fields L/K with Galois group G there is a normal basis, which makes L be free of rank 1 over K[G]. The first results were obtained by Hilbert and Emmy Noether among others, and from the beginning the connection with a condition for the extension called tameness was understood. If we restrict ourselves to abelian number fields, saying that they are tame over Q simply means that they are contained in Q(ζn) with n squarefree. The Hilbert-Speiser theorem states that for abelian number fields tameness is equivalent to the existence of a normal integral basis, namely given by the trace of the n-th root of unity. In general any Galois extension of number fields which has a NIB has to be tame.

Perhaps Noether was the first who understood the importance of looking at the completions: between p-adic fields tameness is a necessary and sufficient condition to have NIB. If we concentrate again on number fields this implies that, just assuming tameness, the structure of OL as an OK[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 an appropriate O_K-algebra, O_K[G] in our case, leads us to the conclusion that L has a NIB over K iff the class of OL in the class group Cl(OK[G]) is trivial; actually there are some subtle questions such as the fact that a priori not all the representatives of the trivial class in Cl(O_K[G]) are free, but we will see that this is the case in our setting. 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 in [GRRS99_{] that Q is the only Hilbert-Speiser number field, i.e. for every} number field K ) Q 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 Cl -Hilbert-Speiser. We can already find some of them in the proof contained in [GRRS99]: the approach of the authors is to find for every number field K an odd prime number

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vi Introduction

l such that K is not Cl-Hilbert-Speiser, noting that this is implied by the fact that the exponent of (O_K/lOK)∗/im(OK∗) does not divide

(l−1)2

2 , where we are taking the image

im of the obvious projection. Then the study of Cl-Hilbert-Speiser fields involved more independent arguments. We will see that Carter and Yoshimura found some criteria for C₂ and C₃-Hilbert-Speiser fields, in [Car03] and [Yos09] respectively, while Herreng, Greither and Johnston in [Her05] and [GJ09] noted that in general Cl-Hilbert-Speiser fields cannot be highly ramified if they are respectively totally imaginary or totally real. We will also see some results by Ichimura.

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 starts in the end of the 50’s from homological algebra and geometric motivations, and it is algebraic K-theory. For example we will speak about the Grothendieck group: the Grothendieck group K₀(R) of a ring R is the free abelian group generated by the finitely generated and projective modules modulo the relationship given by direct sum; we may initially see the class group as the quotient of the Grothendieck group modulo the classes of the free modules. A great part of the earlier results are due to Swan, for instance in [Swa60] and [Swa63]. Another important object is the Whitehead group, which is denoted by K1(R). In the 60’s Milnor noted that from particular commutative

diagrams of rings of the type

Λ Λ1

Λ₂ Λ,

namely fiber products of rings, we can construct an exact sequence of Whitehead and Grothendieck groups, related to the geometric Mayer-Vietoris sequence:

K1(Λ) → K1(Λ1) × K1(Λ2) → K1(Λ) → K0(Λ) → K0(Λ1) × K0(Λ2) → K0(Λ).

From this, under certain hypotheses, Reiner and Ullom in [RU74] concluded that we can restate the Mayer-Vietoris sequence using units and class groups:

0 −→ Λ∗−→ Λ∗₁× Λ∗₂ −→ Λ∗ −→ Cl(Λ) −→ Cl(Λ1) × Cl(Λ2) −→ 0.

This is what is very useful for the proof of [GRRS99]: the augmentation map ε :O_K[G] → OK leads to a fiber product whose Mayer-Vietoris sequence will make the study of Cl(OK[G]) easier, i.e.

OK[G] OK[G]/OKPg∈Gg OK OK/|G|OK,

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vii where we are taking projections and induced maps. We will expose this theory, in its generality, in the third chapter, following the important book by Curtis and Reiner [CR81] and [CR87] on representation theory.

Another important theoretical tool is the study of realizable classes made by McCul-loh in [McC83]. He found a description of the subset of the classes in Cl(O_K[G]) given by the rings of integersO_Lwhen L/K is a tame G-extension, in the case of G elementary abelian (later if G is just abelian, in [McC87]). The first idea is to see Cl(OK[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. Thus we can see McCulloh’s result as a generalization of the classical Stickelberger theorem. 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 of [GRRS99].

In the last chapter we will study weak normal integral bases: we say that L/K has WNIB if the module OL is free after having tensored with the maximal order of

K[G], which is an OK-algebra inside K[G] that contains OK[G]. The weak normal integral bases were studied by Greither in [Gre97] over quadratic fields using an ad hoc McCulloh’s theorem and minus parts of class groups with the help of Iwasawa’s class number formula. With this motivation one may ask whether Q continues to be the only field such that every abelian tame extension has WNIB. The answer seems to be positive, but, if there is any proof, as we will see it is probably going to be hard; what is fundamental for the proof that Q is the only Hilbert-Speiser field is the kernel group, while for this purpose it “disappears” and moves our investigation towards the pure class groups of number fields. Here class field theory is very important: in the second chapter we outline it. After having introduced WNIB’s and spoken about [Gre97] 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 Cl-Leopoldt. The exposed results are not in literature (see also [FG18]) and moreover 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 a mistake contained in the article [Ich16] by Ichimura, 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. 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(ζl) 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.

In the whole exposition we are assuming the basic facts of algebraic number theory, on both global and local fields, Galois theory and commutative algebra. Apart from this, we try to be as self-contained as possible, giving an indication on the parts we are not dealing with.

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Notation

• R[G]: group ring of ring R and group G • RP: localization of the ring R at the prime P

• MP: RP ⊗RM , where M is an R-module and P a prime ideal or a Λ-module for a R-order Λ (in the second case it’s the same of Λ_P ⊗_RM )

• R: completion of the local ring R at its maximal idealb

• M :c R ⊗b _RM , where M is an R-module and R a local ring or a Λ-module for an

R-order Λ (in the second case it is the same ofΛ ⊗b _RM )

• Cl(K): ideal class group of the number field K, namely Cl(OK) • h(K): class number of the number field K

• h(d): class number of Q(√d)

• hn: class number of Q(ζn)

• h(K)+_{: class number of the maximal real field in the CM field K}

• h(K)−_{: relative class number, i.e. h(K)/h(K)}+ _{when K is CM}

• h−_n_{: h(Q(ζ}n))−

• K0(C): Grothendieck group of the category C

• P(R): category of finitely generated projective modules (modulo isomorphisms) • K0(R): Grothendieck group of P(R)

• Cl(Λ): locally free class group of Λ-lattices, where Λ is an R-order • GL(R): direct limit of the invertible matrixes with coefficients in R • E(R): direct limit of the elementary matrixes with coefficients in R

• D(Λ): kernel group of the R-order Λ, i.e. any kernel of the map of extension of scalars to a maximal order contaning Λ

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x Notation • K1(R): Whitehead group of R

• C_l: cyclic group of order l • ∆_l_{: (Z/lZ)}∗

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1 |

Background

1.1. First facts about normal integral bases

In this section we talk about the classical theory of normal integral bases, following mainly [Joh16]. Some proofs will be sketched or omitted.

Let K be a field. It is a well known historical result that, given a finite Galois extension L/K with Galois group G, we can choose an element α ∈ L such that {σ(α) :

σ ∈ G} is a basis of L as a K-vector space: it is the normal basis theorem. It is easy to

see that it can be stated as follows:

Theorem 1.1 (Normal Basis Theorem). Let L/K be a finite Galois extension of fields

with Galois group G. Then L is a K[G]-free module of rank 1.

In algebraic number theory one may ask if this holds true when L and K are number fields or p-adic fields and we consider O_L as a O_K[G]-module. When a finite Galois extension with Galois group G has the property that O_L is free overO_K[G] of rank 1, we say that L/K has a normal integral basis (NIB), i.e. we can easily write the ring of integers of L using the conjugates of an element α ∈O_L with coefficients inO_K.

Remark 1.2. Because of the normal basis theorem, we know that if we have just a free

module it is automatically of rank 1.

Firstly we are interested in finding necessary condition for the extension L/K to have NIB. It will come out that we need the extension to be tame.

Definition 1.3. Let L/K be a finite extension of number fields or p-adic fields, P a prime of L over ℘ of K and p the residue characteristic. We say thatP is tamely ramified in L/K if p - e(P|℘), wildly ramified otherwise. We say that ℘ is tamely ramified in

L/K if every prime of L over ℘ is tamely ramified, wildly ramified otherwise. We say

that the extension L/K is tame if every prime of K is tamely ramified, wild otherwise. An important fact is:

Proposition 1.4. Let L/K be a finite Galois extension of number fields or p-adic fields.

Then it is tame iff Tr_L/KOL=OK.

Now we are ready to prove the following 1

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2 Background Theorem 1.5. Let L/K be a finite Galois extension of number fields or p-adic fields

with Galois group G and NIB generated by α ∈O_L. Then it is tame.

Proof. By the above proposition it is sufficient to prove that Tr_L/KO_L=O_K. We show a stronger result: for every intermediate field F such that F/K is Galois, Tr_L/FOL=OF; let’s define H := Gal(L/F ), so that G/H ∼= Gal(F/K). The fact that TrL/FOL⊆OF is obvious.

Now let x ∈ OF ⊆ OL; then from the NIB property we know that, for certain

aσ ∈OK,

x = X

σ∈G

aσσ(α).

From the fact that ρ(x) = x for every ρ ∈ Gal(L/F ) = H we know that the aσ are invariant in each quotient class, so

x = X σ∈G/H aσσ(TrL/F(α)) = TrL/F   X σ∈G/H aσσ(α)  .

From the proof it is clear that

Corollary 1.6. With the above notation, if L/K has a NIB generated by α then F/K

has a NIB generated by Tr_L/F(α).

The next question one may ask is if tameness could be a sufficient condition. First of all it is much easier to restrict ourselves to abelian extensions, and even the basic Hilbert-Speser theorem fails otherwise. With a view to number fields, we give the following definition:

Definition 1.7. Let K be a number field. If every finite tame abelian extension of K has a normal integral basis we say that K is a Hilbert-Speiser field.

Let C be a class of finite abelian groups. K is C-Hilbert-Speiser if every finite tame abelian extension of K with Galois group in C has a normal integral basis. In particular, if C = {G} we say G-Hilbert-Speiser.

We start from a general fact:

Proposition 1.8. Let L1 and L2 be arithmetically disjoint number or p-adic Galois

extensions over K, embedded in Q. If they have NIB, so does L := L1L2 over K.

Proof. Calling, for i = 1, 2, G_i = Gal(L_i/K) and αi generators of OLi overOK[Gi], we

have that

OL=OL1OL2 ∼=OK[G1]α1⊗OK OK[G2]α2

∼

=OK[G1× G2]α1⊗ α2∼=OK[G]α1α2,

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First facts about normal integral bases 3 With an almost identical proof, we also obtain:

Proposition 1.9. If L1/K has NIB, so does L/L2.

Corollary 1.10. Q(ζn)/Q has NIB if and only if n is squarefree. In that case it is

generated by ζ_n.

Proof. Let n be squarefree (so we can assume it is odd). By Proposition1.8it is sufficient to look at Q(ζp)/Q for p prime, which has a normal integral basis generated by ζp.

If by contradiction n is not squarefree, the extension is not tame. Theorem 1.11 (Hilbert-Speiser theorem). Q is a Hilbert-Speiser field.

Proof. We sketch a proof of this result. First of all we know that every finite abelian

extension L of Q is contained in a cyclotomic field, from the Kronecker-Weber theorem. Let n be the conductor of L, namely the greatest common divisor of m such that

L ⊆ Q(ζm). If L is tame one can prove that n is odd and square-free, and from 1.8 we know that it is a sufficient condition so that Q(ζn)/Q has NIB generated by ζn. So L/Q has NIB by the trace over K of ζn.

Actually Q is the only Hilbert-Speiser field, as we will see later: it is the main result of [GRRS99]. Moreover, the theorem is not true if G is not abelian: in [Mar71] Martinet showed that there exists a tame quaternionic extension of Q without NIB’s. As regards

p-adic fields, we know that the converse holds:

Theorem 1.12. Let L/K be a finite Galois extension of p-adic fields. It is tame if and

only if there is a normal integral basis.

Proof. We sketch a proof. First of all, we consider the case when L/K is totally ramified

with Galois group G of order e coprime to p. It is known for a totally ramified tame extension of p-adic fields that L = K(√e_π

K), for a certain uniformizer πK ∈ K (for example, see [Lan94, §II Proposition 12]); it can be easily proved that we also have OL=OK[e

√

πK]. Now call πL:= e √

πK and consider any element of the type α =Pujπj_L where u_j are units. Using that L/K is a Kummer extension, it can be shown that the matrix of change of basis between {πi_L} and {g(α)}g∈G is a Vandermonde matrix with invertible determinant.

If the extension is unramified, it is sufficient to find a normal basis for the extension of residue fields, that has the same Galois group, and lift it in any way.

In the general case, we call e and f respectively the ramification index and the inertia degree of the extension. By for example [Lan94, §II Proposition 9], there exists a unique unramified extension E/L of degree e. Denoting by F the maximal unramified subextension of E/K, we note that E/K is Galois as E = LF , and by Corollary 1.6

it is sufficient to prove that E/K has NIB. But now the conclusion comes using the result for the extensions E/F and F/K; we have to use a particular NIB generator for

E/F : firstly we construct an uniformizer for E such that π_Ee ∈ K, then we use the NIB generator α = P

πj_E and the auxiliary field K0 = K(πE), that spans E together with

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4 Background

1.2. Algebraic setting

Let K be a number field and G an abelian group. We are interested in what can be said in general about a ring of integers OL over the ring group OK[G] when L is Galois of group G over K, to understand the world where it lives.

We start with some definitions.

Definition 1.13. Let R be a Dedekind domain with field of quotients K, and M an

R-module. We say that M is an R-lattice if it is finitely generated and torsion-free over R. We note that M is embedded in the K-vector space K ⊗RM =: K · M .

Let V be a finite-dimensional K-vector space. M is a full R-lattice in V if V ∼= K ·M . Proposition 1.14. If M and N are two full R-lattices in V , there exists r ∈ R such

that rM ⊆ N .

Proof. Rescaling a pre-assigned basis of V over K, we may suppose it is contained in N . Since M is finitely generated say by m1, ..., mn, for every i the coefficients of mi given by the basis belong to K, and so we obtain r_i ∈ R such that r_imi ∈ N . Taking

r := r1· · · rn we are done.

Definition 1.15. Let R be a Dedekind domain with field of quotients K. An R-order Λ is an R-algebra (there is a homomorphism from R to the centre of Λ) which is an

R-lattice. Λ is embedded in the K-algebra K ⊗RΛ =: K · Λ.

Let A be a finite-dimensional K-algebra. Λ is an R-order in A if A = K · Λ.

Definition 1.16. Let A be a finite-dimensional K-algebra and Λ a R-order in A. M is said to be a Λ-lattice if it is a (left) Λ-module and an R-lattice.

This proposition will be useful in the future:

Proposition 1.17. Let R be a noetherian commutative ring, Λ a noetherian (left)

R-algebra and M a finitely generated (left) Λ-module. Then M is a projective Λ-module if and only if MP is a projective ΛP-module for every P maximal ideal of R.

Proof. One direction is obvious: if M is a direct summand of a free Λ-module, we can

localize.

For the converse, we will use that surjectivity is a local property. So assume that for all P and all surjective homomorphisms N _{L of R-modules we have that}

Hom_Λ_P(M_P, NP) −→ HomΛP(MP, LP)

is a surjection (if M_P is projective, we do know this). Then the analogous global map Hom_Λ(M, N ) −→ Hom_Λ(M, L).

is likewise a surjection, because there is the following isomorphism as RP-modules:

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Algebraic setting 5 (actually it works inverting any moltiplicative subset S of R). This implies that M is projective, establishing the converse implication.

Instead of studying localizations, it is more common to study completions. The next proposition, whose proof can be consulted in [CR81, Proposition 30.17], tells that “it is the same thing”.

Proposition 1.18. Let R be a discrete valuation ring, Λ an R-algebra which is finitely

generated as a R-module and M , N finitely generated left Λ-modules. Then M ∼= N ⇐⇒M ∼c=N ,b

where the first isomorphism is of Λ-modules and the second one of Λ-modules.b

Corollary 1.19. Let R be a noetherian commutative ring, Λ a noetherian (left)

R-algebra and M a finitely generated (left) Λ-module. Then M is locally free in the “local sense” iff it is locally free in the “complete sense”, i.e. M_P is free over Λ_P for every maximal ideal P of R if and only if Md_P is free overdΛ_P for every maximal ideal P of R.

In this case the module M is projective over Λ.

So we can use the term “locally free” without specifying which of the two meanings we are actually using.

Returning to algebraic number theory, with the previous notation we put R =O_K, Λ =O_K[G] and M =O_L. We can now prove the following

Theorem 1.20 (Noether’s theorem). Let L/K be a finite tame Galois extension of

number fields with Galois group G. Then O_L is a locally free O_K[G]-module of rank 1

(thus projective).

Proof. We use Corollary1.19 to allow us to look at completions, proving that for every prime ℘ of O_K, the module O_L,℘ =O_K_℘⊗_O_K O_L is free over O_K[G]℘∼=OK℘[G].

From [Neu99, 8.3] we have that, as K℘-modules, K℘ ⊗K L ∼= LP|℘LP, and an

analogous relationship for the rings of integers. Because of Proposition 1.12 we know that LP/K℘ has NIB for every P|℘, i.e. OL,P is free over OK℘[GP] of rank 1, where

GP = Gal(LP/K℘) is the decomposition group. Call αP0 the generator for a fixed prime

P0.

First, the elements of G act transitively as permutations on the collection of rings OLP. Given an element α in a direct summand of (OL)℘ ∼=

L

P|℘OLP, there exists

g ∈ G such that g−1α ∈OL,P0, and so α can be written as a linear combination of the

elements αP0 with coefficients in O_K

℘[G]. Thus αP0 is a generator of the entire algebra

(OL)℘ ∼=LP|℘OLP.

Actually, overOK[G], the projective modules are exactly the locally free ones. It is a consequence of [Swa60], see [CR81, Theorem 32.11].

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6 Background So now we know that in generalOLbelongs to the class of projectiveOK[G]-module. The NIB question is: when is it actually free? As we will see the question can be restated as the nullity of a certain class in a class group (i.e. a group of modules over O_K[G] modulo an appropriate equivalence relation).

1.3. Representation theory

First of all we briefly outline the general theory of semisimple modules and rings, with sketches of the proofs. Let A be a ring with unit.

Definition 1.21. A non-zero (left) A-module is simple if there are no proper non-zero submodules. It is semisimple if it is a (not necessary direct) sum of simple submodules (eventually infinite).

Lemma 1.22 (Schur’s lemma). If M and N are simple A-modules, then every

non-zero map of HomA(M, N ) is an isomorphism; in that case we have a ring structure

isomorphic to Hom_A(M, M ), and it is a division ring.

Proof. For every element of Hom_A(M, N ) it is sufficient to look at the kernel, and then at the image.

Proposition 1.23. TFAE:

i) M is semisimple;

ii) M is a direct sum of simple submodules;

iii) for every N ⊆ M submodule there exists a submodule P ⊆ M such that M = N ⊕ P . Proof. It can be proved that (ii) ⇒ (i) ⇒ (iii) ⇒ (ii). The first implication is trivial,

while the others are consequences of Zorn’s lemma.

Thanks to the third condition of Proposition 1.23, it is not difficult to prove:

Proposition 1.24. Let A be a ring with identity and writeAA for the left A-module A.

TFAE:

i) every A-module is semisimple; ii) _AA is semisimple;

iii) every A-module is projective.

In this case we will say that A is a semisimple ring.

Remark 1.25. If D is a division algebra, it is simple to see that Mat_n×n(D) is a semisimple ring: it is the direct sum of the column spaces Dn, which are simple as generated by every non-zero column.

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Representation theory 7 Theorem 1.26 (Wedderburn’s theorem). If A is a semisimple ring, then it is a finite

product of matrix rings over division rings.

Proof. Using the module structure ofAA over A and the fact that there is 1 ∈ A,AA is a finite direct sum of simple modules:

AA = M

i

Sai

i

with Si pair-wise non-isomorphic.

As rings A is isomorphic to End_A(_AA)op(opmeans that we define a new multiplication that swaps the original one), thus

A ∼=Y i

EndA(Siai)op∼= Y

i

Matai×ai(EndA(Si))

op∼₌Y i

Matai×ai(EndA(Si)

op₎

and thanks to Schur’s lemma1.22 we obtain the decomposition with division rings.

Proposition 1.27. Let A be a semisimple ring. Then every simple module is isomorphic

to a simple submodule of _AA (which is in fact a minimal left ideal). In particular, there is only a finite number of simple modules over A, up to isomorphism.

Proof. Let M be a simple A-module, and let’s write A = P

Ai with Ai minimal left ideals. So M = AM =P

AiM and we deduce that there exists i such that AiM 6= 0, i.e. there exists m ∈ M with 0 6= Aim ⊆ AiM ⊆ M .

By the simplicity of M we deduce that Aim = M , and using that also Ai is simple we conclude that the multiplication by m induces A_i ∼= M .

We want to study the algebra K[G]. A preliminary observation is the following: Proposition 1.28 (Maschke’s theorem). Let K be any field and G a finite group such

that its order is prime with the characteristic of K. Then K[G] is a semisimple ring, hence separable.

Proof. It is sufficient to prove that any submodule M of K[G] is a direct summand.

Surely M and K[G] are free over K and so there exists a projection p : K[G] → M over

K. Define π : K[G] → M as π(x) := 1 |G| X g∈G gp(g−1x);

it is easy to see that it makes the inclusion M ,→ K[G] split over K[G]. The following definition will be useful:

Definition 1.29. Let K be a field. A separable K-algebra A is an algebra such that, for every extension L/K of fields, L ⊗K A is semisimple. For example, when G is a finite group with order coprime to the characteristic, we may observe that K[G] is even separable.

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8 Background For a moment we focus on the case of K algebraically closed. Here Wedderburn’s Theorem1.26takes an easy form, due to the following remark:

Lemma 1.30. Let K be an algebraically closed field and A a finitely generated division

algebra over K (K has to be contained in the center of A). Then A = K.

Proof. Let α ∈ A. Sice K(α) is commutative, it is a field. Because it is finite as a

K-vector space and K is algebraically closed, we conclude that α ∈ K and so A = K. Corollary 1.31. Let A be a semisimple algebra over an algebraically closed field K.

Then the Wedderburn decomposition of A is

A =Y

i

Mat_a_i×ai(K),

and so every simple component Ai appears with exponent equal to its dimension (they

are of the type Kai_).

In the case of A = K[G], we have that |G| =P

idimK(Ai)2.

The consequences as regards K[G] can be stated with character theory. In the case of G abelian characters are simply the homomorphisms G → K∗ (assuming that K is algebraically closed). It will be very useful in the future to understand what happens in the case of general K and G abelian: here there is an obvious action of GK := Gal( ¯K/K) over the group of characters G∗:= Hom(G, ¯K∗) looking at the image:

γσ(g) := σ(γ(g)) for every γ ∈ G∗, σ ∈ G_K and g ∈ G.

We denote by Φ the set of orbits in G∗. For every γ ∈ G∗ the extension K(γ) of K obtained by adjoining the values of γ is of course a cyclotomic extension of K, and it doesn’t depend on the elements of the same orbit of γ; so we choose for every orbit a character and we prove the following

Theorem 1.32. Let K be a perfect field and G a finite abelian group such that its order

is not divisible by the characteristic of K. Then

K[G] = Y

γ∈Φ

K(γ).

Proof 1. With Wedderburn’s theorem we can identify ¯K[G] and Map(G∗, ¯K), because

the simple modules have dimension 1. More explicitly, for every α ∈ ¯K[G] we consider γ 7→ γ(α) (using a linear extension of the character). On Map(G∗, ¯K) there is an action

of G_K: for every f ∈ Map(G∗, ¯K) put

fσ(γ) := σ(f (γσ−1)).

It is not difficult to see that the action is preserved in the identification. So, taking the fixed parts, K[G] is identified with the maps preserving the action of GK on G∗, which are determined only by the values on some representatives of the orbits. So the statement follows.

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Representation theory 9

Proof 2. The decomposition may be seen through a different approach. Let G = Cn be cyclic of order n, then

K[Cn] ∼= K[z]/(zn− 1),

and the cyclotomic extensions come from the decomposition of zn− 1 in K:

K[Cn] ∼= Y

d|n

(K(ζd) × · · · × K(ζd))

where each factor has a number of copies equal to [Gal(Q(ζd)/Q) : Gal(K(ζd)/K)]. For G general abelian, if we can write G = G₁× G₂, then K[G] ∼= K[G1] ⊗KK[G2]

as K-algebras, and so we can apply the structure theorem.

If K is a number field, one may ask if the order O_K[G] corresponds to QO K(γ) in the above identification. The answer is no: the second term is the maximal order of the algebra K[G], that is the only order which is maximal respect to inclusion; in fact in a commutative algebra an order has to be integral over O_K (it is finitely generated) and of course QO

K(γ) is integral and f.g., so it is the maximal one. In generalOK[G] is not the maximal order:

Proposition 1.33. Let R be a Dedekind domain with field of fractions K and G a group

of order n, where charK - n and n is not invertible in R. Then R[G] is not a maximal order in K[G].

Proof. Let e = 1_nP

g∈Gg /∈ R[G]. As it is central and idempotent, eR[G] + (1 − e)R[G] is an R-order, and it is strictly larger because it contains e which is not in R[G].

Remark 1.34. From the second proof of Theorem 1.32it is clear that the augmentation

ε : K[G] → K, that is the linear map such that g 7→ 1 for every g ∈ G, corresponds to

the evaluation in 1 in K[z] and so to the projection to the trivial component K (this can be less immediately seen also from the first proof of the theorem).

Moreover, the augmentation is split by the inclusion K ,→ K[G], so those compo-nents of K[G] which are not K with trivial action of G span exactly the kernel of the augmentation. The same thing easily holds for the maximal orderQO

K(γ); but we have a splitting even for OK[G] (just with OK ,→ OK[G]), so there are a component that is OK and a kernel component as well.

Now we briefly speak about p-adic character theory. Let Γ be a finite abelian group and p a prime number. We have seen that the ring Qp[Γ] admits a Wedderburn decom-position Qp[Γ] ∼= Y ϕ∈Γ∗ Qp(ϕ). where Γ∗_{= Hom(Γ, Q}p ∗

). By what we are going to see in the next section (Proposition

1.45_{), if p - Γ then Z}_p[Γ] is a maximal order and we get Zp[Γ] ∼=

Y

ϕ∈Γ∗

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10 Background Thus for every character ϕ ∈ Γ∗ _{and a Z}p[Γ]-module M we can consider the compo-nent Mϕ. More explicitly, the ϕ-part of M are the multiples of εϕ, where the εϕ’s are the ortonormal idempotents:

εϕ= 1 |Γ| X g∈Γ ϕ(g)g−1_{∈ Q}p[Γ];

it is easy to verify that ε2_ϕ = ε_ϕ, ε_ϕεϕ0 = 0 for ϕ 6= ϕ0 and 1 =P_ϕ∈Γ∗ε_ϕ, making them

a complete system of ortonormal idempotents and leading to the decomposition

M = X

ϕ∈Γ∗

εϕM.

Moreover, we note that the ϕ-part is the set of elements that are in the ϕ(g)-eigenspace for every automorphism g ∈ Γ.

1.4. Maximal orders

Here we provide a general theory of maximal orders. Almost all of the following results are much simpler in the commutative case, which will be the one we are interested in, but for sake of completeness we expose the general results.

As we already said:

Definition 1.35. Let R be a Dedekind domain with quotient field K, and A a separable finite-dimensional K-algebra. A maximal order is an R-order in A which is maximal with respect to the inclusion.

Definition 1.36. If r ∈ A, then the map a 7→ ra is a K-linear endomorphism of A, and we can consider its characteristic polynomial, which we call p_r. The trace map Tr_A/K(r) is −am−1, where m is the dimension of A over K and pr(x) = xm+ am−1xm−1+ · · · . Lemma 1.37. If charK = 0, the pairing (a, b) 7→ Tr_A/K(ab) is symmetric, associative

and nondegenerate.

Proof. The non-trivial part is to prove that it is nondegenerate. We note that for every

simple component Si of A there exists a finite extension Li/K such that Li ⊗K Si ∼=

M atai×ai(Li). Indeed this is true if we put Li=K, because of lemma 1.30; since under

the identification through K every elementary matrix belongs to F ⊗K Si for a certain finite extension F/K, taking the compositum Li of all such fields we get Li ⊗K Si ∼=

M atai×ai(Li): under the identification for K we have M atai×ai(Li) ,→ Li⊗K Si, but

they have equal dimension over Li.

Now Tr_L_i_⊗_K_S_i_/L_i is nondegenerate, since the set

{a ∈ L_i⊗_KSi∼= M atai×ai(Li) : TrLi⊗KSi/Li(ab) = 0 ∀b ∈ Li⊗KSi}

is a two-sided ideal that is not the whole ring, by our assumption on the characteristic. So it is zero and Tr_L_i_⊗

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Maximal orders 11 Taking the center Ci, that is a field, for every simple component Si of A, we note that Tr_S_i_/C_i is nondegenerate iff Tr_L_i_⊗_K_S_i_/L_i is.

If by contradiction Tr_A/K is degenerate, since A ∼=Q

Si and TrA/K = P

Tr_S

i/K, one

of Tr_S_i_/K is degenerate and so there exists a ∈ Si such that TrSi/K(ab) = 0 for every

b ∈ Si. But then, for every c ∈ Ci and b ∈ Si,

Tr_C_i_/K(Tr_S_i_/C_i(ab) · c) = Tr_C_i_/K(Tr_S_i_/C_i(abc)) = Tr_S_i_/K(abc) = 0,

whence Tr_S_i_/C_i(ab) = 0 for every b ∈ S_i, because Tr_C_i_/K is nondegenerate being C_i/K a

separable extension of fields. This is not possible from what we have said.

Definition 1.38. Let b : A × A → K be a symmetric and nondegenerate K-bilinear form. For every full R-lattice M in A which is free over R with basis {mi}, we define its

discriminant d(M ) to be the ideal of R generated by the determinant of b(m_i, mj). Without the freeness hypothesis, we define the discriminant d(M ) to be the ideal of

R such that every localization is d(MP), as MP is surely free. By Proposition1.14 we can find two free lattices M0 and M00 such that M00⊆ M ⊆ M0. Looking at their bases, we deduce that for all but finitely many maximal ideals P of R we have M_P00 = M_P = M_P0 and d(MP) = RP for almost every P , so the definition is well posed.

Definition 1.39. Let N and M be two full R-lattices in A. Then [M : N ] is the ideal of R such that every localization is [M_P : N_P], namely the fractional ideal generated by the determinant of any map f ∈ Aut(AP = A) such that f (MP) = NP (note that MP and NP are free). As before the definition is well posed.

Lemma 1.40. Let N and M be two full R-lattices in A and b : A × A → K a symmetric

and nondegenerate K-bilinear map; we denote by d its associated discriminant. Then d(M ) = [M : N ]d(N ).

Proof. Straightforward after having localized.

Corollary 1.41. If N ⊆ M , then d(N ) ⊆ d(M ) with equality iff M = N .

Proof. We have to prove that d(N ) = d(M ) implies M = N . It is true locally from the

definition, so M =T

P MP =TP NP = N by [Rei03, Theorem 4.21].

Proposition 1.42. For every R-order Λ there exists a maximal order M such that Λ ⊆ M.

Proof. The complete proof (see [Swa70, Proposition 5.1] or [CR81, Theorem 26.5]) in-volves the concept of reduced trace tr_A/K: the reduced characteristic polynomial of r ∈ A is the product of the characteristic polynomials of the images of a in M atai×ai(L) ∼=

L ⊗KSi, where A ∼=QSi is the decomposition in simple components and L is every field such that Q

M atai×ai(L) ∼= L ⊗KA; then we define the reduced trace as the common

one from the reduced characteristic polynomial, and it is true that in every situation this is nondegenerate and with the relation Tr_A/K = mtr_A/K. The theory of reduced

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12 Background traces and norms can be consulted in chapter 9 of [Rei03]. The important thing is that we have found a symmetric, associative and nondegenerate paring b : A × A → K, which in characteristic zero we know to be the common and simpler trace.

Thus let b : A × A → K a symmetric and nondegenerate K-bilinear map, about which we know the existence, and d the consequent discriminant. If Λ is not maximal nor contained in a maximal ideal, there exists an infinite sequence

Λ ( Λ1 ( Λ2( · · · , and therefore an infinite ascending sequence of ideals in R

d(Λ) ( d(Λ1) ( d(Λ2) ( · · · .

Because R is noetherian, this is a contradiction.

Remark 1.43. If A is commutative, then there is only one maximal order, that is the

integral closure M of R in A. In fact every element of an order is integral over R, because lies in a finitely generated R-algebra. That M spans A is clear looking at any polynomial that annihilates every element of A. It remains to show that M is finitely generated: in the case of K[G] with K number field it will be evident, but also the general proof is a standard argument involving the nondegenerate bilinear map and Cramer’s rule (see [CR81, Proposition 26.7]).

Corollary 1.44. An R-order Λ is maximal if and only if every localization ΛP is.

Proof. Let M ⊇ Λ be a maximal order. If every ΛP is maximal, then ΛP = MP and Λ = M by Corollary1.41.

For the converse, it is true that for every multiplicatively closed subset of R, if Λ is maximal, also ΛS is. The proof is not difficult and can be consulted in [Swa70, Theorem 5.28].

Now everything is clear for group rings. Together with Proposition1.33, we get:

Proposition 1.45. Let G be a finite group of order n and R[G] te group ring. Then

the primes of R at which the localization of R[G] is not a maximal order are those in the factorization of n in R. So, by the above corollary, R[G] is a maximal order if and only if n is invertible in R.

Proof. It is sufficient to prove that every order that contain R[G] is contained in n−1R[G].

So let x =P

g∈Gagg ∈ Λ and h ∈ G: we want to show that nah ∈ R. Since xh−1 ∈ Λ, it is simple to verify that its trace Tr_K[G]/K(xh−1) belongs to R. By linearity and the fact that if g ∈ G then Tr_K[G]/K(g) is n for g = e and 0 otherwise, we conclude.

The following fact is non-trivial, and follows from the characteristic of maximal orders to be hereditary, i.e. every one-sided ideal is projective. We will see that it is true for Dedekind domain, and so easily for the maximal order of K[G] when K is a number field and G a finite abelian group.

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Maximal orders 13 Theorem 1.46. If Λ is a maximal order in a separable algebra A, R is a discrete

valuation ring with quotient field K and M and N are finitely generated torsion free

Λ-modules, then M ∼= N if and only if K ⊗RM ∼= K ⊗RN .

Proof. See [Rei03, Theorem 18.10] or [Swa70, Theorem 5.27] (the latter uses Grothendieck groups and the factorization of ideals in maximal orders).

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2 |

Class field theory

In this chapter we outline the principal results related to class field theory. The main philosophy behind it is to describe the abelian extensions of a number field or local field K using only information we can get looking at K. The first development happened between the end of the 19th century and the Second World War, with protagonists Kronecker, Weber, Hilbert, Takagi, Artin and Hasse. The proofs involved analytic arguments, while in the following few decades the theory has been reformulated and proved again with idélic and cohomological tools, thanks to Chevalley and Tate among others. A complete amount of the statements and proofs for this second approach is given in Milne’s notes [Milne13] (which also consider some analytic points of view). Neukirch’s book [Neu99] is a good reference as well. Moreover we add some considerations about minus parts of class groups, that are more or less related to class field and will be very useful for the study of weak normal integral bases.

2.1. Global class field theory

Let K be a number field and let S be a finite set of prime ideals in K. We denote by

I_KS = IS the subgroup of the group of fractional ideals generated by the primes which are not in S. Let KS be the set of elements a ∈ K∗ such that aO_K ∈ IS_{, Cl(K) the} ideal class group of K and U :=O_K∗. The following proposition, whose straightforward proof can be read in [Milne13, §V Lemma 1.1] (the main tool is the Chinese remainder theorem), states a relationship between these objects.

Proposition 2.1. For every finite set of primes S, the following is exact: 0 −→ U −→ KS −→ IS −→ Cl(K) −→ 0,

where the maps are the natural ones.

Now it is important to consider a larger definition of prime.

Definition 2.2. Let K be a number field. We define non-archimedean primes as the maximal ideals in OK and archimedean primes as the real embeddings K → R or the pairs of complex-conjugate embeddings K → C. These are called the primes or places of K.

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16 Class field theory For the archimedean primes we can define ramification as well. First of all, an extension of a real archimedean prime v : K → C to L, where L/K is an extension of algebraic extensions of Q (not necessarily number fields), is an extension of the map v to w : L → C (there is always one by Zorn’s lemma), and if imaginary it has to be considered with its conjugate; similarly, if (v, v) is a pair of complex embedding of K, an extension is of the type (w,w) with w extension of v. If L/K is Galois, we note

that Gal(L/K) acts transitively on the extensions of a fixed archimedean prime of K. Thus we can define a decomposition/inertia subgroup of Gal(L/K) given an extension of archimedean places w|v (simplified notation for w|v, (w,w)|v or (w, w)|(v, v)):

D(w|v) = E(w|v) : = {σ ∈ Gal(L/K) : wσ = w}

= {σ ∈ Gal(L/K) : w ◦ σ = w ∧ w ◦ σ = w}.

Generalizing a possible definition of ramification index for non-archimedean primes, for any (not necessary Galois) extension L/K with primes w|v, the ramification index is e(w|v) := [E(u|v) : E(u|w)], where u is an extension of w to Q. Therefore we may simply define the ramification index as follows:

Definition 2.3. Let w|v be an extension of archimedean places in L/K (not necessarily Galois). Then their ramification index is e(w|v) = 2 if v is real and w is complex, and 1 otherwise.

Now we continue with class field theory.

Definition 2.4. Let V be the set of all primes of K (both archimedean and non-archimedean). A modulus m is a map m : V → N such that its support is finite and does not contain complex primes and is 0 or 1 on real primes. We shall write

m = Y

℘∈V

℘m(℘)= m0m∞,

where m0 is a product of (not necessarily distinct) non-archimedean primes and m∞ a

product of real primes with multiplicity at most 1. We will say that a prime ℘ divides

m if m(℘) ≥ 1.

We can define a generalization of the ideal class group that considers a given modulus

m:

Definition 2.5. Let K be a number field and m a modulus. We define the ray class

group modulo m by the quotient

Cl_m(K) := IS(m)/ι(Pm),

where S(m) is the set of primes that appear in m₀ and P_m is the set of a ∈ K∗ such that a ≡ 1 (mod ℘m(℘)) for every non-archimedean prime ℘ (note that there is only a finite number of conditions) and ℘(a) > 0 for every real embedding ℘ : K → R with

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Global class field theory 17 Of course the classical ideal class group is the ray class group modulo 1. There is an important relation between any ray class group and the ideal class group; we start with the following

Lemma 2.6. Let K be a number field and m a modulus. Then the quotient KS(m)/Pm

is finite and isomorphic to:

KS(m)/Pm ∼= Y ℘ real ℘|m {±1} × Y ℘ non-archimedean ℘|m (OK/℘m(℘))∗ ∼= Y ℘ real ℘|m {±1} × (OK/m0)∗.

Proof. We sketch the proof, which can be found in [Milne13, §V Theorem 1.7]. The image of a ∈ KS(m) in the real components is simply the sign of ℘(a). A standard argument that uses the Chinese remainder theorem (e.g. [Milne13, §V Remark 1.2; Lemma 1.5]) shows that we can write a = b/c with b, c ∈ O_K ∩ KS(m)_{; then the projection of a in} OK/℘m(℘) is simply the image of b/c, as c is invertible.

Theorem 2.7. Let K be a number field and m a module. Then the following is exact: 0 −→ U/Um−→ KS(m)/Pm −→ Clm(K) −→ Cl(K) −→ 0,

where Um := U ∩ Pm. In particular every group in the exact sequence is finite and,

defining h_m := |Cl_m(K)| and h = h_K the class number, we have hm= h 1 [U : U_m]2 |℘|m real|_{N (m} 0) Y ℘ non-archimedean ℘|m 1 − 1 N (℘) ,

where N is the absolute norm.

Proof. The proof at this point is quite straightforward and done in [Milne13, §V Theorem 1.7].

Corollary 2.8. If hK = 1 and m = m0 is an integral ideal, then

Cl_m(K) = (O_K/m)∗/im(O_K∗),

where the image comes from the natural map.

Before going ahead, we give some definitions.

Definition 2.9. Let L/K be a Galois extension of number fields. Given a prime ℘ of

K unramified in L, for every prime P of L over ℘ there exists a Frobenius element

(P, L/K) ∈ G that generates the decomposition group, namely the unique σ ∈ G such that

σ(α) ≡ αN (℘) (mod ℘)

for every α ∈ O_K, where N is the absolute norm; by (℘, L/K) ⊆ G we denote the subset of Frobenius elements as P changes over ℘, which are conjugate to each other via the automorphisms of L/K (so if the extension is abelian we have just one Frobenius automorphism, that depends only on the prime of K).

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18 Class field theory Definition 2.10. Let L/K be an abelian extension of number fields and let S be a finite set of non-archimedean primes of K that contains all the ramified in L ones. We define the global Artin map to be the map that extends multiplicatively the Frobenius one:

ψ_L/K: IS −→ Gal(L/K) ℘n1 1 · · · ℘ nt t 7−→ t Y i=1 (℘_i, L/K)ni_.

Definition 2.11. Let L/K be an extension of number fields. For every extension of prime ideals P|℘ we define the norm of P as N_L/K(P) = ℘f (P|℘)_{, and we extend it} multiplicatively to every fractional ideal. It is the norm map N_L/K. We can verify that it is also defined between ideal class groups (on principal ideals, i.e. elements, it is the usual norm map: it is sufficient to consider the normal closure of L over K); we continue to call it N_L/K.

The following proposition will be useful in the future. The proof is a simple hand-made verification, and this situation is known as capitulation.

Proposition 2.12. Let L/K be an extension of number fields. Then the composition

of the extension map E : Cl(K) → Cl(L) with the norm N_L/K : Cl(L) → Cl(K) is the

multiplication by the degree of the extension: N_L/K ◦ E = [L : K]. In particular, the

kernel of the extension E is annihilated by [L : K].

Now we are ready to state the main theorems of global class field theory.

Theorem 2.13 (Reciprocity law). Let L/K be an abelian extension of number fields

and let S be the set of non-archimedean primes of K ramifying in L. Then there exists a modulus m (we say that L/K admits the modulus m) such that S(m) = S, m∞ is the

product of the ramified archimedean primes and the Artin map ψ_L/K : IS → Gal(L/K)

passes to the quotient for the ray class group, i.e. ψ_L/K(ι(P_m)) = {identity}. Moreover

it defines an isomorphism

I_KS(m)/ι(Pm) · NL/K(ILS(m)) −→ Gal(L/K),

where N_L/K is the restriction of the norm map and, with an abuse of notation, we denote by S(m) also the primes of L lying over the ones of K belonging to S(m).

The same property holds for every modulus divisible by m.

Definition 2.14. The smallest such modulus (under divisibility) is called conductor

f (L/K) of the extension. We have such a minimal one thanks to Theorem 2.7: after having taken any suitable modulus m, for every ℘|m non-archimedean prime, according to Lemma 2.6there is the smallest integer f (℘) ≤ m(℘) such that the following is well defined:

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Global class field theory 19 and moreover the modulus f = Q

℘f (℘)m∞ makes the Artin map pass to the quotient

Cl_f(K) → Gal(L/K). f is the conductor and, as for m, the primes that divide it are exactly the ramified ones.

The theorem that puts together a large class of finite abelian extensions of a number field is the following

Theorem 2.15 (Existence theorem). Let K be a number field and m a modulus. Then

for every group H such that ι(P_m) ⊆ H ⊆ I_KS(m) (such groups are in correspondence with the subgroups of Clm(K)) there exists a finite abelian extension L/K such that

H = ι(Pm) · NL/K(I S(m)

L ), and so I S(m)

K /H ∼= Gal(L/K). Of course one can expect a good behavior for containments:

Theorem 2.16. Let L1/K and L2/K be abelian extensions of conductor dividing a

modulus m. Then L1 ⊆ L2 ⇐⇒ ι(Pm) · NL1/K(I S(m) L1 ) ⊇ι(Pm) · NL2/K(I S(m) L2 ) ι(Pm) · NL1L2/K(I S(m) L1L2) =ι(Pm) · NL1/K(I S(m) L1 ) ∩ι(Pm) · NL2/K(I S(m) L2 ) ι(Pm) · NL1∩L2/K(I S(m) L1∩L2) =ι(Pm) · NL1/K(I S(m) L1 ) ·ι(Pm) · NL2/K(I S(m) L2 ) .

Definition 2.17. If in the existence theorem we set H = ι(Pm), we obtain the ray class

field modulo m, which we denote by K(m). It has the property that Gal(Km/K) ∼= Clm(K) and every field with conductor dividing m is contained in it.

Remark 2.18. If m = 1 the ray class field is called Hilbert class field, and it is the maximal

unramified (also at archimedean primes) abelian extension of K.

Example 2.19. Let K = Q and m = n∞ where n ∈ N and ∞ is the only real prime of Q. Then IS(m)is the group of ideals generated by rational numbers r such that (r, n) = 1,

while P_m are the ones generated by a positive rational number r such that r ≡ 1 (mod n); hence the ray class field Cln∞(K) ∼= (Z/nZ)∗.

For L = Q(ζn) we know explicitly the Artin map: it is simple to verify that the

Frobenius automorphism of a prime p ∈ Z is ζn 7→ ζnp. So we can check that the

extension Q(ζn)/Q admits the modulus n∞. By cardinality we conclude that Q(ζn) is

the ray class field modulo n∞.

If we simply took the modulus to be n, in a similar way we get that Q(ζn)+ := Q(ζn+ ζn−1) is the ray class field modulo n.

Corollary 2.20 (Kronecker-Weber theorem). Let K be an abelian extension of Q with

conductor f = f0f∞. Then it is contained in the f0-th cyclotomic field.

We return to Hilbert class fields. As they are the ray class fields modulo 1, the Hilbert class field HK of K is such that Gal(HK/K) ∼= Cl(K). With this isomorphism there is an identification between norm maps and restrictions of automorphisms:

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20 Class field theory Proposition 2.21. Let L/K be an extension of number fields. The extension LHK/L is unramified (every inertia subgroup goes into the one of H_K/K) and so LHK ⊆ HL.

Therefore we have a map Gal(H_L/L) → Gal(HK/K), that is the composition of the

re-striction Gal(HL/L) → Gal(LHK/L) with the immersion Gal(LHK/L) ,→ Gal(HK/K);

overall it is a restriction to H_K. Then the following is commutative:

Cl(L) Gal(H_L/L)

Cl(K) Gal(HK/K), ψ_HL/L

NL/K restriction

ψ_{HK /K}

where the horizontal maps are the isomorphisms given by the Artin maps.

Proof. We can prove it starting from a prime ideal P of L that lies over ℘ in K. Through

the Artin map, it goes to the element σP ∈ Gal(HL/L) such that

σP(α) ≡ αN (P) (mod P)

for every α ∈O_H_L and anyP over P. Therefore

σP|HK(α) ≡ (α

N (℘)₎f (P|℘) _(mod_{P ∩ O} HK)

for every α ∈O_H_K; it is the image through the other Artin map of ℘f (P|℘), that is the norm of P.

Corollary 2.22. Let L/K be an extension of number fields with no unramified abelian

subextensions different from K. Then the norm map Cl(L) → Cl(K) is surjective (and so h_K | h_L).

Proof. The hypothesis tells us that L ∩ H_K= K, whence the restriction on the right of the commutative diagram is surjective.

Remark 2.23. To show divisibility of class numbers, one can also work directly with

Galois groups: the quotient hL/hK is the order of Gal(LHK/L).

An important consequence of the statements of class field theory, together with some analytic considerations, is Chebotarev theorem.

Definition 2.24. Let T be a set of primes in a number field K. If there exists 0 ≤

δ(T ) ≤ 1 such that X ℘∈T 1 N_K/Q(℘s₎ ∼ δ(T ) log 1 s − 1

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Minus parts 21 Theorem 2.25 (Chebotarev theorem). Let L/K be a finite Galois extension of number

fields with Galois group G and let C be a subset of G stable under conjugation.

Then the set T of primes ℘ in K unramified in L such that (℘, L/K) ⊆ C has positive Dirichlet density δ(T ) = 1/[G : C].

The proof can be consulted in [Milne13, §VIII Theorem 7.14].

2.2. Minus parts

Every number field K has a certain number r_K of real embeddings and s_K pairs of conjugate complex embeddings. As we said before, rK is the number of real primes and

sKis the number of imaginary primes. Of course we have the relation [K : Q] = rK+2sK, and we define a field K as totally real or totally imaginary if respectively s_K = 0 or

rK= 0. A Galois extension of Q is either totally real or totally imaginary (in the second case its degree over Q is even).

Definition 2.26. A number field K is CM if it is totally imaginary and there is a totally real subfield K+ such that [K : K+] = 2.

Remark 2.27. One can verify that complex conjugation is a well defined automorphism

of a CM-field K (i.e. it is independent from the embedding in C). For instance, see [Was97] just before Theorem 4.10. Moreover, the restriction of a conjugation to another CM-field is still the conjugation, so we can speak in general about the conjugation map. Definition 2.28. Let K be a CM-field and K+_{its real subfield. The extension K/K}+_is

ramified at the archimedean primes, so by Corollary2.22h+_K := h_K+ | h_K. The quotient

will be called the relative class number h−_K:= h_K/h+_K.

It has an algebraic meaning. When K/K0 is a quadratic extension of number fields with Galois group ∆ = {1, j}, we would like to define a minus part of the class group of K using j. Firstly, as we will see from the next proof, we have to take the odd part of it (namely the product of the cyclic factors of order a power of an odd prime, or the subgroup of elements of odd order), to have a Zh12

i

[∆]-module structure: we define Cl(K)− := Ker ((1 + j) : Cl(K)odd −→ Cl(K)odd)

and we consider the restriction of the extension map E : Cl(K0)odd −→ Cl(K)odd (of course the odd part goes to the odd part); it is injective, because by Proposition 2.12

the kernel, of odd order, is annihilated by 2. Theorem 2.29. Under the above hypotheses,

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22 Class field theory

Proof. Defining Cl(K)+:= Ker ((1 − j) : Cl(K)odd → Cl(K)odd), as 1+j₂ and 1−j₂ form a complete system of ortonormal idempotents we can write Cl(K)_odd = Cl(K)+⊕ Cl(K)− (we used the ring Zh1₂i[∆] as a support for our study of the group structure).

It is now sufficient to prove that ImE = Cl(K)+. As it is the extension map, it is clear that ImE ⊆ Cl(K)+. Moreover, if x ∈ Cl(K)+ we can write x =1₂x1+j and we can represent 1₂x as an ideal I of Cl(K) (which lives in the odd part), and so x = [I]1+j

comes from Cl(K0).

Corollary 2.30. If K is a CM-field, the odd part of h−_K is the order of Cl(K)−.

We have already noted that for a CM field K the extension map E : Cl(K+) → Cl(K) has kernel annihilated by 2. Actually the kernel has order 1 or 2, as proved in [Was97, Theorem 10.3]. It is 1 for cyclotomic fields, see [Was97, Theorem 4.14].

The following result will be applied when we discuss weak normal bases.

From Stickelberger’s theorem (see for example [Was97, Theorem 6.10], or the end of the presentation of McCulloh’s theorem for prime cyclotomic fields) we know that the Stickelberger ideal annihilates the class group of an abelian field. In the case of cyclotomic fields of prime power type, Iwasawa [Iwa62] gave a more explicit relationship between the class number and the index of the Stickelberger ideal; a proof can be also read in [Was97, Theorem 6.19].

First of all he considered minus parts of the group ring:

Z[∆]−:= Ker ((1 + σ−1) : Z[∆] −→ Z[∆]) = Im ((1 − σ−1) : Z[∆] −→ Z[∆])

where p is an odd prime, n ≥ 1 and ∆ = Gal(Q(ζpn)/Q) = {σ_i: (i, pn) = 1}. We denote

by J the Stickelberger ideal of Z[∆] and J− := J ∩ Z[∆]−. Theorem 2.31. In the above notation, [Z[∆]− : J−] = h−

Q(ζpn).

We finish with a result involving extensions of CM-fields. First of all, in an extension

L/K of CM-fields the norm map is well defined also on the minus parts, and we denote

it by N_L/K− : Cl−(L) → Cl−(K). In fact the norm map between ideal class groups commutes with the conjugation j: for every pair of primes P|℘ of L/K we have

NL/K(P)j = ℘f (P|℘)j= (℘j)f (P|℘) = (℘j)f (P

j_|℘j₎

= N_L/K(Pj);

moreover, j commutes with every immersion of L over K for the same reason for which the conjugation is independent from the immersion in C of the CM-fields. So if [I]1+j = 0, i.e. I1+j = (x), then

N_L/K([I])1+j = N_L/K([I]1+j) = [N_L/K(x)] = 0.

Proposition 2.32. Let L/K be an odd extension of CM-fields. Then the norm N_L/K− : Cl−(L) → Cl−(K) is surjective. In particular the odd part of h−_K divides the odd part of h−_L.

On Hilbert-Speiser and Leopoldt fields

Universit`

a degli Studi di Pisa

On Hilbert-Speiser and Leopoldt fields

Contents

Introduction

Notation

1

|

Background

1.1. First facts about normal integral bases

1.2. Algebraic setting

1.3. Representation theory

1.4. Maximal orders

2

|

Class field theory

2.1. Global class field theory

2.2. Minus parts