Content uploaded by David Brink

Author content

All content in this area was uploaded by David Brink on Oct 29, 2015

Content may be subject to copyright.

Procyclic Galois Extensions of Algebraic

Number Fields

David Brink

Ph.D. thesis

Defence:

February 23, 2006

Thesis advisor:

Christian U. Jensen, Københavns Universitet, Denmark

Evaluating committee:

Ian Kiming, Københavns Universitet, Denmark

Niels Lauritzen, Aarhus Universitet, Denmark

Noriko Yui, Queen’s University, Canada

Matematisk Afdeling ·Københavns Universitet ·2006

David Brink

Matematisk Afdeling

Universitetsparken 5

2100 København Ø

Denmark

E-mail: brink@math.ku.dk

c

2006 David Brink (according to the Danish legislation)

Contents

Preface 5

Abstract 6

1 Iwasawa’s theory of Zp-extensions 9

1.1 Introduction and notation . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Ramiﬁcation in Zp-extensions .................... 10

1.3 Rank and essential rank of pro-p-groups .............. 10

1.4 Leopoldt’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 The maximal number of linearly disjoint Zp-extensions . . . . . . 13

1.6 The dihedral Iwasawa number . . . . . . . . . . . . . . . . . . . . 15

1.7 Hyperprimary elements . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 A theorem of Shafarevich . . . . . . . . . . . . . . . . . . . . . . . 17

1.9 The notion of p-rationality...................... 19

1.10 Prime decomposition in ring class ﬁelds . . . . . . . . . . . . . . . 20

2 On Zp-embeddability of cyclic p-class ﬁelds 23

2.1 Introduction.............................. 23

2.2 Criteria for p-rationality ....................... 24

2.3 Algorithm to determine Zp-embeddability . . . . . . . . . . . . . 26

3 Prime decomposition in the anti-cyclotomic extension 31

3.1 Introduction.............................. 31

3.2 The cyclotomic and the anti-cyclotomic extension . . . . . . . . . 32

3.3 Prime decomposition laws . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Discriminants with one class per genus . . . . . . . . . . . . . . . 43

3.5 The ﬁrst step of the anti-cyclotomic extension . . . . . . . . . . . 45

4 The ring class ﬁeld of conductor 2∞over imaginary quadratic

number ﬁelds 50

4.1 Introduction.............................. 50

3

4.2 The ring class ﬁeld of conductor 2∞and the anti-cyclotomic extension 51

4.3 The genus ﬁeld and other elementary abelian extensions of K. . 53

4.4 Prime decomposition in the anti-cyclotomic extension . . . . . . . 54

4.5 The norm form of the eighth cyclotomic ﬁeld . . . . . . . . . . . . 57

4.6 The form class group of discriminant −4lfor a prime l≡1 (mod 4) 59

4.7 The form class group of discriminant −8lfor an odd prime l. . . 61

4.8 The form class group of discriminant −ll0with two primes l,l0. . 63

4.9 The ﬁelds Q(√−l) with an odd prime l............... 63

4.10 The ﬁelds Q(√−2l) with an odd prime l.............. 68

4.11 The ﬁelds Q(√−ll0) with two primes l≡1 (mod 4) and l0≡3 (mod 4) 72

4.12 Embeddability of 2-class ﬁelds into Z2-extensions . . . . . . . . . 75

4.13 Interrelations between Q(√−l) and Q(√−2l) ........... 77

4.14Numericalexamples.......................... 79

5 Non-abelian ﬁbre products as Galois groups 83

5.1 Introduction: rank, socle, ﬁbre product . . . . . . . . . . . . . . . 83

5.2 The p-adic prodihedral groups Dp.................. 84

5.3 Realising ﬁbre products of D2with itself . . . . . . . . . . . . . . 85

5.4 D2-extensions with diﬀerent base, but common socle . . . . . . . . 86

5.5 Realising the ﬁbre product of Z2×Z2and D2........... 91

Bibliography 94

Preface

The present booklet constitutes my Ph.-D. thesis in mathematics. It was written

in the period 2003–2005 under the supervision of Christian U. Jensen whom it is

my pleasure to thank warmly for his interest and support over the years.

The thesis consists of 5 chapters. Chapter 1 is an introduction to the theory

of procyclic Galois extensions. Chapters 2 and 3 are extended versions of my

papers [3] and [4]. Chapters 4 and 5 are based on two papers still in preparation.

For the beneﬁt of the busy reader, I have included a thorough abstract to the

entire thesis.

5

Abstract

Denote by Zpthe additive group of p-adic integers. The main theme of this thesis

is the existence and properties of Galois extensions of algebraic number ﬁelds with

Galois group Zp, in short Zp-extensions. We shall however also consider some non-

abelian pro-p-groups as Galois groups (in particular in Chapter 5). The thesis is

divided into 5 chapters.

Chapter 1. In this introductory chapter, results of Iwasawa and Shafarevich are

summarised, note in particular Theorems 1, 3, and 6. The connection between

Zp-extensions and Leopoldt’s Conjecture is discussed. The notion of p-rationality

is deﬁned, and the classiﬁcation of 2-rational imaginary quadratic ﬁelds is given

(Theorem 10), apparently for the ﬁrst time (correctly).

Chapter 2. Consider an imaginary quadratic number ﬁeld Kand an odd prime

p. The following question is investigated: if the p-class group of Kis non-trivial

and cyclic, is the p-Hilbert class ﬁeld of K(or part of it) then embeddable in a

Zp-extension of K? It is shown that the answer is always yes when Kis p-rational

(Lemma 12), and two criteria for p-rationality are given (Theorem 13 of which

only part (b) is new). Some examples are given indicating that “most” Kare p-

rational. When Kis not p-rational, an eﬀective algoritm is given that determines

Zp-embeddability (Theorem 15). Numerical examples show that all ﬁve cases of

that theorem occur.

Chapter 3. Consider an imaginary quadratic number ﬁeld Kand an odd prime

p. The (well-known) fact that Khas a unique Zp-extension which is prodihedral

over Qis shown (Proposition 18). We call this extension the anti-cyclotomic

p-extension of K.

We give laws for the decomposition of prime ideals in the anti-cyclotomic

extension (Theorems 22 and 24). The laws involve representation of some rational

prime power qhby certain quadratic forms of the same discriminant dKas K.

Using Gauss’ theory of composition of forms, we show that it suﬃces instead to

represent qby some form (Section 3.3). The whole story becomes particularly

simple when each genus of forms of discriminant dKconsists of a single class

6

(Theorem 28). This happens for 65 values of dKclosely connected to Euler’s

numeri idonei or convenient numbers

The decomposition laws also depend on how many steps of the anti-cyclotomic

extension are unramiﬁed. This dependence may be turned around, meaning that

if we know how certain primes decompose, then we can compute the number of

unramiﬁed steps (Examples 25–27). In particular, we can answer whether the

p-Hilbert class ﬁeld of Kis contained in the anti-cyclotomic extension and thus

is Zp-embeddable.

In section 3.5 we show how to ﬁnd explicit polynomials whose roots generate

the ﬁrst step of the anti-cyclotomic extension. When Kis not p-rational this

involves using the decomposition laws to identify the right polynomial famong

a ﬁnite number of candidates (Examples 31 and 32). When this is done, one

obtains nice laws for the splitting of fmodulo q. For instance we show that

X5+ 5X2+ 3 splits into linear factor modulo a prime number q6= 3,5 if and only

if qis of the form x2+ 5xy + 100y2or 3x2+ 15xy + 50y2.

Chapter 4. Consider an imaginary quadratic number ﬁeld Kwith Hilbert class

ﬁeld KH. The ring class ﬁeld N=N(2∞) over Kof conductor 2∞is the maximal

2-ramiﬁed (i.e. unramiﬁed outside 2) abelian extension of Kwhich is generalised

dihedral over Q.

We determine the structure of the Galois group Gal(N/KH) (Lemma 33) and,

in some cases, Gal(N/K) (Corollaries 39, 44, 49).

We give a law for the decomposition of prime ideals in the anti-cyclotomic

2-extension of K(Theorem 34) similar to that from Chapter 3 (but more com-

plicated). Again, this law involves representation of rational prime powers by

quadratic forms.

In Sections 4.5–4.8, quadratic forms are discussed. For example, we show

the new result (Lemma 36) that a prime number congruent to 1 modulo 16 is

representable by both or none of the forms X2+ 32Y2and X2+ 64Y2, whereas

a prime number congruent to 9 modulo 16 is representable by one, but not both

of these forms. New proofs of two formulae of Hasse regarding the order of cyclic

2-class groups are given. The key ingredient in these proofs are two new explicit

expressions ((4.10) and (4.12)) for a form representing a class of order 4 in the

form class group.

The ﬁrst step of the anti-cyclotomic 2-extension Kanti/K is of the form K(√a)

with an a∈Z. As a Z2-extension, Kanti/K is unramiﬁed outside 2. However, the

lower steps might be unramiﬁed also at 2. Let νdenote the number of unramiﬁed

steps. When the 2-class group of Kis cyclic (possibly trivial), we give algorithms

to compute both νand a(Theorems 37, 42, 47). In most cases we can even

7

give explicit expressions for ν(Theorems 38, 43, 48) and a(Theorems 41, 46, 50

of which 41 and 46 are not new). The proofs of these results involve the class

number formulae of Hasse.

When the 2-class ﬁeld of Kis non-trivial and cyclic, one can ask if it can

be embedded into a Z2-extension of K. We answer this question completely

(Theorem 51) using many of our previous results. For any (odd or even) prime p,

it is conjectured that there exist imaginary quadratic ﬁelds with Zp-embeddable

p-class ﬁeld of arbitrarily high degree,

Put K=Q(√−l) and K0=Q(√−2l) with a prime l≡1 (mod 8). There

are some quite surprising interrelations between these two ﬁelds. Let hand h0

be the class numbers of Kand K0, respectively. We show 8 |h⇔8|h0for

l≡1 (mod 16), and 8 |h⇔8-h0for l≡9 (mod 16) (Theorem 52). We

also give results on interrelations between the anti-cyclotomic 2-extensions of K

and K0(Theorems 54, 55). Finally, a conjecture for primes l≡1 (mod 16) is

put forth that would allow a certain assumption in Theorems 54 and 55 to be

omitted. This conjecture has been veriﬁed by the author for all primes up to 14

millions.

In the last section, many numerical examples are given showing that the results

of the previous sections are best possible.

Chapter 5. The pro-2-group H=ha, b |ba =a−1b−1, ba−1=ab−1iis described

as a ﬁbre product of two copies of the 2-adic prodihedral group D2. The socle of a

H-extension M/Qis deﬁned as its unique biquadratic subﬁeld. It is investigated

which biquadratic extensions of Qappear as socle of a H-extension. This is for

example the case for Q(√−1,√2) (Example 58). If the socle of a H-extension

M/Qis of the type Q(√−l, √2) with an odd prime l, it is shown than Mcontains

a square root of either √2 + 2 or √2−2 (Lemma 59). The determination of the

right square root is not trivial, and some partial results in this direction are given

(Theorems 60 and 61).

The pro-2-group G=ha, b |ab2=b2a, a2b=ba2iis described as a ﬁbre

product of Z2×Z2with D2.Gis not realisable as Galois group over Q. Some

results on the number ν(G, K) of G-extensions of imaginary quadratic ﬁelds of

type K=Q(√−l) or K=Q(√−2l) with lan odd prime are given, for example

it is shown that always ν(G, K)≤3 (Theorem 63). Further, it is shown that the

free pro-2-group of rank 2 is realisable over Kin some cases (l≡3,5 (mod 8)),

but not in others (l= 353, for example).

8

Chapter 1

Iwasawa’s theory of Zp-extensions

1.1 Introduction and notation

Let pbe a ﬁxed prime number. Denote by Zpthe additive group of p-adic integers.

We have

Zp∼

=lim

←− Z/pn,

i.e. Zpis the inﬁnite procyclic pro-p-group.

We shall consider Galois extensions of algebraic number ﬁelds with Galois

group Zp, in short Zp-extensions.

In this chapter, an overview of important results is given. In particular we

emphasise Theorem 1 and Theorem 3 which are due to Iwasawa [18], Theorem 6

(and its Corollary 7) due to Shafarevich [26] , and the deﬁnition of the anti-

cyclotomic extension at the end of section 1.6.

In all of this chapter 1, we use the following notation:

p: a prime number

Zp: the additive group of p-adic integers

ζ: a primitive p’th root of unity

K: an algebraic number ﬁeld

O: the ring of integral elements in K

E: the group of units O

r1, r2: the number of real and complex primes of K, respectively

Up: the group of local units at p(note Up=K∗

pfor pinﬁnite)

KH: the Hilbert class ﬁeld of K

K∞: the maximal abelian p-extension of Kunramiﬁed outside p

KZp: the composite of all Zp-extensions of K

9

1.2 Ramiﬁcation in Zp-extensions

An algebraic extension L/K is called unramiﬁed outside pif all primes pof

Kwith p-p(including the inﬁnite ones) are unramiﬁed. More generally, for a

(usually ﬁnite) set Sof primes of K, it is said that L/K is unramiﬁed outside

Sif all primes p6∈ Sare unramiﬁed. We shall later see that being unramiﬁed

outside a ﬁnite set of primes is a rather strict condition.

Theorem 1. Any Zp-extension Lof the algebraic number ﬁeld Kis unramiﬁed

outside p.

Proof. Let pbe a prime of Kwith p-pand assume indirectly that pramiﬁes

in L. Consider a localisation Lp/Kp. This means the following: Pick some prime

Pof Lextending p. Let Ln⊂Lbe the subextension of degree pnover Kand

denote by Ln,pthe completion of Lnwith respect to the restriction of Pto Ln.

Then we have the tower

KpjL1,pjL2,pj. . .

and Lpis deﬁned as the union of the Ln,p.

We may assume that Lp/Kpis a totally ramiﬁed Zp-extension. By the below

valuation-theoretic lemma, each Ln,p/Kpis a radical extension. Hence Kpcontains

a primitive pn’th root of unity for all n, a contradiction.

Lemma 2. Let Fbe a ﬁeld with a complete and discrete valuation. Assume

E/F is a totally and tamely ramiﬁed extension of ﬁnite degree n. Then there

exists a uniformising element Π∈Ewith Πn∈F. In particular E/F is a radical

extension.

Proof. Let Π and πbe uniformising elements for Eand F, respectively. Write

Πn=uπ with some unit u∈E. Since Fand Ehave the same residue ﬁeld, we

may pick a unit u0∈Fwith u0≡u(mod Π). Put u∗:= u/u0and π∗:= u0π.

Then Πn=u∗π∗where π∗∈F, and the unit u∗∈Esatisﬁes u∗≡1 (mod Π).

By Hensel’s Lemma, u∗is an n’th power: u∗=vn. The uniformising element

Π∗:= Π/v then satisﬁes (Π∗)n=π∗. Now replace Π with Π∗.

1.3 Rank and essential rank of pro-p-groups

Let Xbe a pro-p-group. The Frattini subgroup Φ(X) of Xis the closed

subgroup generated by the commutators and the p’th powers. The quotient

X/Φ(X) is an elementary abelian p-group. The rank of Xis deﬁned as the

10

dimension of X/Φ(X) as vectorspace over Fp. By Burnside’s Basis Theorem1,

this rank equals the cardinality of any minimal generating subset of X.

Now let Xbe an abelian pro-p-group. We may view Xas a (compact) Zp-

module. Iwasawa deﬁnes the essential rank of Xas the dimension over the

p-adic numbers Qpof the tensor product

X⊗ZpQp.

If Xhas ﬁnite rank, the Elementary Divisor Theorem gives

X∼

=(Zp)a×T

where a < ∞is the essential rank of X, and Tis a ﬁnite p-group.

Let us ﬁnally note that if

1→X→Y→Z→1

is an exact sequence of abelian pro-p-groups, then exactness is conserved by ten-

soring with Qp, and hence

ess.rank(Y) = ess.rank(X) + ess.rank(Z).

In particular, ess.rank(X) = ess.rank(Y) if Zis ﬁnite.

The reason we introduce the above concepts is this: We shall take as Xthe

Galois group Gal(K∞/K). It will then be a key point of this chapter to show that

rank(X) is ﬁnite and to give an expression for this rank as well as for ess.rank(X).

We can interpret the essential rank as the maximal number of linearly disjoint

Zp-extensions of K.

1.4 Leopoldt’s conjecture

Consider the algebraic number ﬁeld Kand the rational prime p. For any prime p

of Kdividing p, denote by Upthe group of local units at p, i.e. the group of units

in the ring of integers Opof the completion Kp. Further, consider the higher unit

groups

U(n)

p={x∈Up|x≡1 (mod pn)}

for n≥1.

1Burnside’s Basis Theorem is well known for ﬁnite p-groups, see for instance [17]. It can be

extended to pro-p-groups without too much trouble. Note incidentally that the cyclic group of

order 6 shows that the assumption that Xis a p-group can not be omitted.

11

For suﬃciently large n, the p-adic logarithm is an isomorphism

logp:U(n)

p→(pn,+).

Hence U(n)

pis a free Zp-module of rank [Kp:Qp]. It follows that U(1)

pis a Zp-

module of rank [Kp:Qp]. Hence the direct product

U(1) := Y

p|p

U(1)

p

is a Zp-module of rank

X

p|p

[Kp:Qp] = [K:Q].

Let Ebe the group of global units of Kand put

E1={∈E| ∀p|p:≡1 (mod p)}.

The abelian group E1is a subgroup of Eof ﬁnite index and hence has the same

rank which is

rank(E1) = rank(E) = r1+r2−1

by Dirichlet’s unit theorem (r1and r2have the usual meaning). We may consider

E1as a subgroup of U(1) via the embedding

E1→U(1), 7→ (,...,).

The closure E1of E1with respect to the topology of U(1) is a (compact) Zp-

module.

One could think that the Zp-rank of E1equals the Z-rank of E1. This might

also very well be true, however only the inequality

Zp-rank(E1)≤r1+r2−1

is clear. We have the

Leopoldt Conjecture for the field Kand the prime p: The Zp-rank

of E1is r1+r2−1.

This conjecture was formulated by Leopoldt [21] in 1962 for totally real ﬁelds.

If K=Qor Kis imaginary quadratic, then r1+r2−1 = 0 and hence

the Leopoldt Conjecture is trivially true. Further, it was shown by Brumer [6]

that the Leopoldt Conjecture is true if Kis abelian over Qor over an imaginary

quadratic ﬁeld. In the general case, however, neither proof nor counter-example

is known.

12

1.5 The maximal number of linearly disjoint Zp-

extensions

Let K∞be the maximal abelian p-extension of Kwhich is unramiﬁed outside p.

By Theorem 1, any Zp-extension of Kis contained in K∞. Write

[K:Q] = r1+ 2r2

where r1and 2r2are the numbers of real and complex embeddings of K, respec-

tively.

Theorem 3. The rank of the abelian pro-p-group Gal(K∞/K)is ﬁnite, and the

essential rank asatisﬁes

r2+ 1 ≤a≤[K:Q].

Equality a=r2+ 1 holds if and only if the Leopoldt Conjecture for the ﬁeld K

and the prime pis true.

We have

Gal(K∞/K)∼

=(Zp)a×T

with a ﬁnite p-group T.

Proof. We start by summarising some results from class ﬁeld theory. Let Jbe

K’s id`ele group. For any abelian extension L/K, the global norm symbol

(, L/K) : J→Gal(L/K)

is a continuous, surjective homomorphism. The kernel Nis called the normgroup

of L. The mapping L7→ Ngives a 1-1 corresponding between the abelian exten-

sions L/K and the closed subgroups NjJcontaining the principal id`eles K∗

and with J/N totally disconnected. Moreover, a prime pof Kis unramiﬁed in

L/K iﬀ the normgroup Ncontains the group Upof local units at p.

Put

U0=Y

p|p

Up, U00 =Y

p-p

Up, U =U0×U00.

Uis an open subgroup of J. The normgroup of the maximal abelian extension

of Kunramiﬁed outside pis

H=U00K∗.

So Gal(K∞/K) is isomorphic to the p-part of J/H. The normgroup corresponding

to K’s Hilbert class ﬁeld is H0=U K∗. Clearly

H⊂H0⊂J,

13

and H0has ﬁnite index in J(because J/H0is isomorphic to K’s class group).

Evidently U0H=H0, so

H0/H ∼

=U0/(U0∩H).

Let U(1) be as in section 1.4. It is a subgroup of U0of ﬁnite index. So U(1)/(U(1) ∩

H) has ﬁnite index in U0/(U0∩H).

Recall we have an embedding

ψ:E1→U(1).

This embedding does not commute with the standard embedding K∗→J, so we

cannot omit the ψhere.

We claim

ψ(E1) = U(1) ∩H.

For an ε∈E1, we have

ψ(ε) = ε·ψ(ε)

ε∈K∗U00

and hence E1jK∗U00 =H. Proving the other inclusion is somewhat technical

and we omit the details. The reader is referred to [27], page 266.

Write the rank of E1(as a Zp-module) as r1+r2−1−δwith a δ≥0. Then δ= 0

iﬀ the Leopoldt Conjecture for Kand pholds. Hence U(1)/(U(1) ∩H) = U(1) /E1

has Zp-rank

[K:Q]−(r1+r2−1−δ) = r2+1+δ

by section 1.4. This module is isomorphic to a submodule of J/H of ﬁnite index.

It follows that the p-part of J/H has ﬁnite rank and that its essential rank is

r2+1+δ. The claims follow by section 1.3.

The composite KZpof all Zp-extensions of K(inside a ﬁxed algebraic closure)

is called the maximal Zp-power extension of K. By Theorem 3, Gal(KZp/K)

is a free Zp-module of rank a. Therefore, a=a(K) is the maximal number of

linearly disjoint Zp-extensions of K. No number ﬁeld Kis known for which a(K)

depends on the prime p(hence the notation); in fact no Kis known for which

a(K)6=r2+ 1 since that would constitute a counter-example to the Leopoldt

Conjecture.

For an arbitrary (abstract) ﬁeld k, it still holds that Gal(kZp/k) is a free

Zp-module, but its rank is in general no longer equal to the essential rank of

Gal(k∞/k) (see [12]). The rank of Gal(kZp/k) is called the Iwasawa number

of kwith respect to p. For number ﬁelds, we henceforth use the term Iwasawa

number instead of the equivalent essential rank.

14

From Theorem 3 follows immediately a(Q) = 1, i.e. there is a unique Zp-

extension of Qfor any p. We can describe this extension explicitly. Adjoint to

Qall roots of unity of p-power order. By class ﬁeld theory, this is the maximal

abelian extension of Qunramiﬁed outside {p, ∞}. Its Galois group over Qis

isomorphic to Zp×Z/(p−1) for p > 2 and to Z2×Z/2 for p= 2. Hence it has a

unique subﬁeld Qcycl with Galois group Zpover Q. We call Qcycl the cyclotomic

Zp-extension of Q. In the simplest case p= 2, one ﬁnds

Q⊂Q√2⊂Qq2 + √2⊂ ··· ⊂ Qcycl .

For any number ﬁeld K, the composite Kcycl =KQcycl is a Zp-extension of K

called the cyclotomic extension of K.

From Theorem 3 also follows a(K) = 2 for an imaginary quadratic ﬁeld K.

Hence Khas maximally 2 linearly disjoint Zp-extensions. One such, of course, is

Kcycl. We shall have more to say on ﬁnding a “complementary” Zp-extension of

K.

In chapter 2, we shall concern ourselves with the determination of the torsion

Tfrom Theorem 3 in case Kis imaginary quadratic. A ﬁrst step is to compute the

rank of Gal(K∞/K). This is done in section 1.8 using a theorem of Shafarevich.

To formulate and prove this result, we ﬁrst need to introduce the concept of

hyperprimary elements in section 1.7.

1.6 The dihedral Iwasawa number

For a prime p, deﬁne the p-adic prodihedral group Dpas the natural projective

limit of the dihedral groups of order 2pn,n≥1:

Dp= lim

←− Dpn.

Dpcontains the procyclic group Zpas unique abelian subgroup of index 2. Any

element τ∈Dp\Zphas order 2 and inverts Zpby conjugation. So we may write

Dpas the semidirect product

Dp=Zpo Z/2.

If a ﬁeld extension M/K has Gal(M/K)∼

=Dp, we denote the subﬁeld corre-

sponding to the subgroup Zpas the quadratic base of the Dp-extension.

Now let L/K be a quadratic extension of number ﬁelds and consider the

maximal Zp-power extension LZpfor some prime p. Let a(K) and a(L) be the

Iwasawa numbers of Kand Lwith respect to p. Deﬁne L+and L−as the maximal

15

subextensions of LZp/L, normal over K, such that Gal(L/K) operates trivially

on Gal(L+/L) and by inversion on Gal(L−/L), respectively. Then LZpis the

composite of L+=KZpLand L−and hence

Gal(L+/L)∼

=Za(K)

p,Gal(L−/L)∼

=Za(L)−a(K)

p

(see section 3 of [12] for details on this).

We call a(L/K) := a(L)−a(K) the dihedral Iwasawa number of K(with

respect to p). It is the maximal number of linearly disjoint (over L)Dl-extensions

with quadratic base L/K.

Clearly, L+and L−are linearly disjoint over Lfor p > 2, but it is not always

the case for p= 2. This will cause us some trouble.

Now consider an imaginary quadratic ﬁeld K. It has dihedral Iwasawa number

a(K/Q) = a(K)−a(Q) = 2 −1 = 1.

Hence there exists a unique Dp-extension with quadratic base K/Qfor every

prime p. We call it the anti-cyclotomic Zp-extension of Kand denote it Kanti.

As noted previously, Kcycl and Kanti are linearly disjoint over Kwhen p > 2. For

p= 2, however, the intersection could be K(√2) which is always the ﬁrst step of

the cyclotomic 2-extension of K.

1.7 Hyperprimary elements

Consider a ﬁnite set Sof primes of K. We shall mainly be interested in the

case S={p|pdivides p}, but for the moment Sis arbitrary. Deﬁne the set of

hyperprimary elements

V:= {x∈K∗|(x) = apfor an ideal ajO}

and the set of S-hyperprimary elements

VS:= V∩\

p∈S

Kp

p.

Evidently one has V=V∅and the inclusions

K∗pjVSjVjK∗.

The quotient VS/K∗pis a vectorspace over Fp, the dimension of which is denoted

σ(S). First we compute σ(∅):

16

Lemma 4. Let E=O∗be the group of units in Kand let Cbe the class group

of K. Then

dim(V/K∗p) = rankp(E) + rankp(C).

Proof. For a hyperprimary x∈V, the ideal awith (x) = apis unique. Therefore

V−→ C, x 7→ [a]

is a well-deﬁned homomorphism. The image is {[a]∈C|[a]p= 1}, and the

kernel is E·K∗p. The lemma follows.

Remark 5. Assume that Kcontains a primitive p’th root of unity and that

Scontains all primes dividing p. Then there is the following characterisation of

S-hyperprimary elements:

x∈VS⇔

In the extension K(p

√x)/K, every ﬁnite

prime (i.e. every prime ideal) is unramiﬁed,

and moreover, every p∈Ssplits.

In this case, K(p

√VS) is the maximal elementary abelian p-extension of Kin

which all prime ideals are unramiﬁed and all p∈Ssplit. Kummer theory then

gives

VS/K∗p∼

=Gal(K(p

pVS)/K).

A reference to hyperprimary elements is [16].

1.8 A theorem of Shafarevich

Let Sbe a ﬁnite set of primes of the number ﬁeld K. Deﬁne KSas the maximal

elementary abelian p-extension of Kwhich is unramiﬁed outside S, and let d(S)

be the dimension of Gal(KS/K) over Fp. In other words, d(S) is the maximal

number of linearly disjoint Z/p-extensions of Kunramiﬁed outside S. The fol-

lowing theorem of Shafarevich links this number to the dimension σ(S) of VS/K∗p

(see section 1.7).

Theorem 6. Let t(S)be the number of non-complex primes p∈Ssuch that

the completion Kpcontains a primitive p’th root of unity ζ. Put δ= 1 if ζ∈K,

else δ= 0. Further, let

λ(S) = X

p∈S, p|p

[Kp:Qp]

17

and r=r1+r2−1. Then one has the equality

d(S) = σ(S) + t(S)−δ+λ(S)−r .

In particular, KSis a ﬁnite extension of K.

Proof. The proof is somwhat similar to that of Theorem 3 whose notation we

reuse. The deﬁnition of KSgives that its normgroup is the open group

NS=USJpK∗

with US=Qp6∈SUp. So we have to compute the dimension of

J/NS∼

=Gal(KS/K).

Consider the following sequence of vectorspaces over Fp:

1−→ VS/K∗pf4

−→ V/K∗pf3

−→ U∅/USUpf2

−→ J/NS

f1

−→ J/N∅−→ 1.

Here f1,f2and f4are deﬁned the natural way. For an x∈V, the principal id`ele

(x) is the product of a u∈U∅and a yp∈Jp. The id`ele uis unique modulo Up.

Therefore f3:x7→ ugives a well-deﬁned homomorphism.

With a little work, it is seen that the above sequence is exact. We show below

that all dimensions are ﬁnite. Thus

dim(VS/K∗p)−dim(V/K∗p)+ dim(U∅/USUp)−dim(J/NS)+dim(J/N∅) = 0.(∗)

The dimension of VS/K∗pis σ(S) by deﬁnition. The dimensionen of J/NSis d(S)

by deﬁnition. We have

U∅/USUp∼

=Y

p∈S

Up/Up

p,dim(U∅/USUp) = X

p∈S

dim(Up/Up

p).

By the determination of powers in valued ﬁelds (use Hensel’s lemma or see [16],

for instance), U/USUphas dimension t(S) + λ(S). Note that K∅is the maximal

unramiﬁed elementary abelian p-extension of K. Hence dim(J/N∅) equals the p-

rank of K’s class group C. By Dirichlet’s unit theorem, the p-rank of K’s group of

units Eis r+δ. Lemma 4 now gives that V /K∗phas dimension r+δ+ dim(J/N∅).

Putting everything into (∗) gives the claim.

It is an important, but straightforward observation that d(S) equals the

rank of the Galois group over Kof the maximal p-extension unramiﬁed out-

side S(see deﬁnition of rank in section 1.3). The same goes for the maxi-

mal abelian p-extension of Kunramiﬁed outside S. In particular we get for

18

S={p|pdivides p}:

Corollary 7. The rank of the pro-p-group Gal(K∞/K)is σ(S)+t(S)−δ+r2+1.

This rank can be computed explicitly in some simple cases:

Corollary 8. Assume p= 2 and that K=Q(√−∆) is imaginary quadratic

with a square-free ∆∈N. Then the rank of the pro-2-group Gal(K∞/K)is

(r+ 2 if all pi≡ ±1 (mod 8)

r+ 1 otherwise

where ris the number of odd primes p1, . . . , prdividing ∆.

Proof. Clearly r2=δ= 1. Let Sbe the set of primes of Kdividing 2. So

t(S) is the cardinality of S, i.e. t(S) = 2 when ∆ ≡ −1 (mod 8) so that 2 splits

in K, else t(S) = 1. To compute σ(S), let E/K be the maximal unramiﬁed

abelian 2-extension in which all primes p∈Ssplit. So σ(S) is the 2-rank of

Gal(E/K ). The genus ﬁeld Fof Kcontains E. Let µbe the number of primes

dividing K’s discriminant dK. So µequals r+ 1 or raccording to whether dK

is even or odd. Deﬁne p∗

i:= ±pisuch that p∗

i≡1 (mod 4). Genus theory gives

F=K(√p∗

1,...,√p∗

r). The 2-rank of Gal(F/K) is µ−1. The degree of F/E is

1 or 2 since this extension is cyclic (Gal(F/E) is the decomposition group of an

unramiﬁed extension). Now note for a p∈S:

psplits in K(pp∗

i)/K ⇔p∗

iis a square in Kp=Q2(√−∆)

⇔p∗

ior −∆p∗

iis a square in Q2

⇔p∗

ior −∆p∗

iis ≡1 (mod 8)

It follows that σ(S) = µ−1 if all pi≡ ±1 (mod 8) or −∆≡5 (mod 8), else

σ(S) = µ−2. Putting everything into Corollary 7 gives the claim.

1.9 The notion of p-rationality

The situation is particularly simple when

Gal(K∞/K)∼

=(Zp)r2+1.

In this case Kis called p-rational. This notion was introduced in [19].

19

Theorem 9. The following conditions are equivalent:

(a) The ﬁeld Kis p-rational.

(b) Gal(K∞/K)has rank r2+ 1.

(c) Gal(K∞/K)is torsion-free, and the Leopoldt Conjecture for Kand pholds.

(d) One has VS=K∗pwhere VSdenotes the set of S-hyperprimary elements

in Kfor S={p|pdivides p}. Further, if Kcontains a primitive p’th root

of unity ζ, then Khas only one prime pdividing p. If Kdoes not contain

ζ, then neither do the completions Kpwith p|p.

Proof. The equivalence of (a), (b), and (c) follows immediately from Theorem 3.

The equivalence of (b) and (d) follows from Corollary 7.

Classifying the p-rational ﬁelds is not trivial. We show here one result in that

direction and return to the question in section 2.2.

Theorem 10. (a) Qis p-rational for all primes p.

(b) The 2-rational imaginary quadratic number ﬁelds are exactly Q(√−1),

Q(√−2),Q(√−l), and Q(√−2l)for primes l≡3,5 (mod 8).

Proof. (a) Let Q(ζp∞) denote the ﬁeld obtained by adjoining to Qall roots of

unity of p-power order. Then the maximal abelian extension of Qunramiﬁed

outside pis the maximal real subﬁeld of Q(ζp∞). This is a Z2-extension for p= 2

and a Zp×Z/((p−1)/2)-extension for p > 2. The claim follows. (One also sees

that Qis 2-rational by noting that Q(√2) is the only quadratic extension of Q

which is unramiﬁed outside 2.)

(b) For an imaginary quadratic ﬁeld Kwe have r1= 1, so p-rationality means

that Gal(K∞/K) has rank 2. The claim now follows from Corollary 8.

Note that the classiﬁcation of the 2-rational imaginary quadratic ﬁelds in [19]

(Corollaire 1.3) is not correct.

1.10 Prime decomposition in ring class ﬁelds

One of the central results in class ﬁeld theory is the law on decomposition of

prime ideals in abelian extensions L/K of algebraic number ﬁelds. We consider

here the case where Kis imaginary quadratic and L/Qis a generalised dihedral

extension:

Gal(L/Q)∼

=Gal(L/K)o Z/2.

20

Then Lis contained in a ring class ﬁeld N(f) over Kwith suitable conductor f

due to a theorem of Bruckner (see [5]).

The ring class group over Kof conductor fis the group IK(f) of fractional

K-ideals prime to fmodulo the subgroup PK(f) generated by the principal ideals

(α) with integral α≡a(mod f) for some a∈Zprime to f. By class ﬁeld theory,

there is a canonical isomorphism (the Artin symbol)

IK(f)/PK(f)→Gal(N(f)/K).

To the ﬁeld Lcorresponds a subgroup of Gal(N(f)/K) which again by the Artin

symbol corresponds to a subgroup H0of IK(f). For a prime ideal pof Kprime

to f, we now have the following decomposition law: psplits in Lif and only if

p∈H0(see [22], Theorem 7.3).

Now consider the group Cof classes of forms of discriminant dKf2where dK

is the discriminant of K. There is a canonical isomorphism between the ring class

group IK(f)/PK(f) and the form class group C(see [8], Theorem 7.7 and 7.22).

Let Hbe the subgroup of Ccorresponding to H0under this isomorphism. Then

a prime number pis representable by some form (class) f∈Hif and only if pis

the norm of an ideal p∈H0([8], Theor