ThesisPDF Available

Procyclic Galois Extensions of Algebraic Number Fields

Authors:

Abstract

Denote by $Z_p$ the 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 $Z_p$, in short $Z_p$-extensions. We shall however also consider some non-abelian pro-$p$-groups as Galois groups.
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 2over imaginary quadratic
number fields 50
4.1 Introduction.............................. 50
3
4.2 The ring class field of conductor 2and 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 l1 (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 l1 (mod 4) and l03 (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 2is 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 aZ. 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 l1 (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 |h8|h0for
l1 (mod 16), and 8 |h8-h0for l9 (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 l1 (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 =a1b1, ba1=ab1iis 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 22 (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 (l3,5 (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 LnLbe 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 ΠEwith ΠnF. In particular E/F is a radical
extension.
Proof. Let Π and πbe uniformising elements for Eand F, respectively. Write
Πn=with some unit uE. Since Fand Ehave the same residue field, we
may pick a unit u0Fwith u0u(mod Π). Put u:= u/u0and π:= u0π.
Then Πn=uπwhere πF, and the unit uEsatisfies u1 (mod Π).
By Hensel’s Lemma, uis 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
XZpQp.
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
1XYZ1
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={xUp|x1 (mod pn)}
for n1.
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+r21
by Dirichlet’s unit theorem (r1and r2have the usual meaning). We may consider
E1as a subgroup of U(1) via the embedding
E1U(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+r21
is clear. We have the
Leopoldt Conjecture for the field Kand the prime p: The Zp-rank
of E1is r1+r21.
This conjecture was formulated by Leopoldt [21] in 1962 for totally real fields.
If K=Qor Kis imaginary quadratic, then r1+r21 = 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 Kbe 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) : JGal(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
HH0J,
13
and H0has finite index in J(because J/H0is isomorphic to K’s class group).
Evidently U0H=H0, so
H0/H
=U0/(U0H).
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/(U0H).
Recall we have an embedding
ψ:E1U(1).
This embedding does not commute with the standard embedding KJ, so we
cannot omit the ψhere.
We claim
ψ(E1) = U(1) H.
For an εE1, we have
ψ(ε) = ε·ψ(ε)
εKU00
and hence E1jKU00 =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+r21δwith a δ0. 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+r21δ) = 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/(p1) 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
QQ2Qq2 + 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,n1:
Dp= lim
Dpn.
Dpcontains the procyclic group Zpas unique abelian subgroup of index 2. Any
element τDp\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 Las 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 Land 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 Lare 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:= {xK|(x) = apfor an ideal ajO}
and the set of S-hyperprimary elements
VS:= V\
pS
Kp
p.
Evidently one has V=Vand the inclusions
KpjVSjVjK.
The quotient VS/Kpis a vectorspace over Fp, the dimension of which is denoted
σ(S). First we compute σ():
16
Lemma 4. Let E=Obe the group of units in Kand let Cbe the class group
of K. Then
dim(V/Kp) = rankp(E) + rankp(C).
Proof. For a hyperprimary xV, the ideal awith (x) = apis unique. Therefore
VC, x 7→ [a]
is a well-defined homomorphism. The image is {[a]C|[a]p= 1}, and the
kernel is E·Kp. 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:
xVS
In the extension K(p
x)/K, every finite
prime (i.e. every prime ideal) is unramified,
and moreover, every pSsplits.
In this case, K(p
VS) is the maximal elementary abelian p-extension of Kin
which all prime ideals are unramified and all pSsplit. Kummer theory then
gives
VS/Kp
=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/Kp
(see section 1.7).
Theorem 6. Let t(S)be the number of non-complex primes pSsuch that
the completion Kpcontains a primitive p’th root of unity ζ. Put δ= 1 if ζK,
else δ= 0. Further, let
λ(S) = X
pS, p|p
[Kp:Qp]
17
and r=r1+r21. 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:
1VS/Kpf4
V/Kpf3
U/USUpf2
J/NS
f1
J/N1.
Here f1,f2and f4are defined the natural way. For an xV, the principal id`ele
(x) is the product of a uUand a ypJp. 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/Kp)dim(V/Kp)+ dim(U/USUp)dim(J/NS)+dim(J/N) = 0.()
The dimension of VS/Kpis σ(S) by definition. The dimensionen of J/NSis d(S)
by definition. We have
U/USUp
=Y
pS
Up/Up
p,dim(U/USUp) = X
pS
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 Kis 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 /Kphas 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 pSsplit. 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
i1 (mod 4). Genus theory gives
F=K(p
1,...,p
r). The 2-rank of Gal(F/K) is µ1. 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 pS:
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) = µ1 if all pi≡ ±1 (mod 8) or 5 (mod 8), else
σ(S) = µ2. 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=Kpwhere 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 l3,5 (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/((p1)/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 αa(mod f) for some aZprime 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
pH0(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) fHif and only if pis
the norm of an ideal pH0([8], Theorem 7.7). There follows:
Proposition 11. Consider an imaginary quadratic field Kwith discriminant
dK. Let Lbe an abelian extension of Kcontained in K’s ring class field N(f)
of conductor f. Denote by Cthe form class group of discriminant dKf2. Let
Hbe the subgroup of Ccorresponding to Lunder the canonical isomorphism
Gal(N(f)/K)
=C. Let pbe a prime number dividing neither dKnor f. Then p
splits totally in the generalised dihedral extension L/Qif and only if pis repre-
sentable by a form in H.
Remark. Antoniadis [1] gives another criterion for the splitting of primes in ring
class fields: For each character ψon Gal(N(f)/K) with ψ26= 1, he considers the
L-series
L(ψ, s) =
X
n=1
aψ(n)ns
where ψis viewed as a character on the absolute Galois group of K. This L-series
coincides with the Artin L-series
L(IndQ
K(ψ), s)
coming from a dihedral type Galois representation over Q. Antoniadis then shows
that a prime p-fsplits in N(f) if and only if the p’th coefficient satifies aψ(p) = 2
21
for all ψ(Satz 2). A main result in Antoniadis’ article is the explicit determination
of all the coefficients aψ(n) (page 204) from which Proposition 11 can be deduced.
22
Chapter 2
On Zp-embeddability of cyclic
p-class fields
2.1 Introduction
Let pbe an odd prime and consider an imaginary quadratic number field K. As
shown by Iwasawa (Theorem 1), any Zp-extension of Kis unramified outside p.
The lower steps of a such extension might well be unramified also at p. In this
chapter 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? In doing so, we are led to study the torsion subgroup
of the Galois group over Kof the maximal abelian p-extension of Kwhich is
unramified outside p. First fix some notation:
p: an odd prime number
ζ: a primitive p’th root of unity
∆ : a square-free natural number
K: the imaginary quadratic number field Q(∆)
O: the ring of integral elements in K
K0: the p-Hilbert class field of K
Ke: the p-part of K’s ray class field with conductor pe,e=0
K: the union S
e=0 Ke
T: the torsion subgroup of Gal(K/K)
Kcycl : the cyclotomic Zp-extension of K
Kanti : the anti-cyclotomic Zp-extension of K
I: the group of fractional ideals of Kprime to p
P: the group of principal fractional ideals of Kprime to p
Pe: the ray modulo pe,e0
23
Note that the ray class field with conductor 1 is exactly the Hilbert class field so
that the notation is consistent. We have the tower
KjK0jK1jK2j··· ⊂ K
and note that the union Kis the maximal abelian p-extension of Kwhich
is unramified outside p. Thus, by Iwasawa’s result, any Zp-extension of Kis
contained in K. It is well known that Kis the composite of three fields Kcycl,
Kanti, and KTwhich are linearly disjoint over K(see chapter 1). The cyclotomic
extension Kcycl is the unique Zp-extension of Kwhich is abelian over Q. The
anti-cyclotomic extension Kanti is the unique Zp-extension of Kwhich is pro-
dihedral over Q. Finally, KTis a finite extension of Kwith Gal(KT/K)
=Tand
dihedral over Q(but not unique with these properties). As we shall see, we may
usually for KTtake K0or a subfield of K0. From the above discussion follows
the isomorphism
Gal(K/K)
=Zp×Zp×T
which will be important in the following. It may also be noted that the composite
KantiKTis the maximal abelian p-extension of Kwhich is unramified outside p
and dihedral over Q, and hence equals the union of all p-ring class fields over K
with conductor a power of p.
2.2 Criteria for p-rationality
The concept of p-rationality has an obvious connection to the question of Zp-
embeddability:
Lemma 12. Let p > 2and assume that the imaginary quadratic field Kis p-
rational. Then the p-Hilbert class field K0of Kis cyclic (possibly trivial) and
embeddable into an Zp-extension of K.
Proof. Since K0is dihedral over Q, it is contained in KantiKT. So if Kis p-
rational and the torsion Tthus trivial, it will follow that K0is contained in Kanti.
We remark that the situation is more difficult for p= 2. Here a cyclic 2-class
field can be Z2-embeddable even though it is not contained in Kanti. We return
to this problem in section 4.12.
What we need now are criteria for p-rationality.
Theorem 13. (a) Q(3) is 3-rational. Let K=Q(∆) with a square-free
24
N. Then Kis 3-rational if 6≡ 3 (mod 9) and the class number of Q(3∆)
is not divisible by 3.
(b) Assume p5and let Kbe as above. Then Kis p-rational if it has the
same p-class number as K(ζ)where ζis a primitive p’th root of unity.
Proof. (a) We use condition (d) of Theorem 9.
K=Q(3) contains a primitive third root of unity ζ, and 3 ramifies in K.
If Khad a non-trivial hyperprimary element xVS\K3where Sis the set of
primes dividing 3, then K(3
x)/K would be an unramified Z/3-extension. Since
Khas class number 1, this is impossible. Thus Kis 3-rational.
Now let K=Q(∆) with ∆ 6= 3. Then Kdoes not contain ζ. We have
the biimplication
ζKp=Q3(∆) 3 (mod 9) .
Let K0be the 3-Hilbert class field of K. By a theorem of Kubota (see [20]), the
p-class number of K(3) is the product of the p-class numbers of K,Q(3),
and Q(3∆) for an odd prime p. It then follows from the assumption and the
fact that Q(3) has trivial class number that K(3) has the same 3-class
number as K. Hence K0(3) is the 3-Hilbert class field of K(3). Clearly
K0(3)/K is abelian. Assume for a contradiction that Khas a non-trivial hy-
perprimary element xVS\K3. Then K(3,3
x)/K(3) is an unramified
Z/3-extension (see Remark 5). Therefore 3
xis contained in the 3-Hilbert class
field K0(3). But K(3
x) is not normal over K, a contradiction. Hence Kis
3-rational.
(b) In the case p5 neither Knor Kp=Qp(∆) contain ζ. The same
argument as above shows VS=Kpwhere Snow is the set of primes dividing p.
Remarks. (a) We shall later see that Kis p-rational for p5 when its class
number is not divisible by p.
(b) Note that part (b) of Theorem 10 generalises the similar part of [19],
Corollaire 1.3.
(c) The proof of Theorem 10 shows that it suffices to assume in part (b) that
the p-class groups of Kand K(ζ) have the same ranks. However this seems to
happen only when the p-class numbers are also identical.
(d) Theorem 10 never applies when pis an irregular prime, i.e. when the class
number of Q(ζ) is divisible by p.
The below table shows all values of ∆ <200 for which the 3-part hof the class
25
number of K=Q(∆) is divisible by 3. An asterix means that ∆ 3 (mod 9).
h0is the 3-part of the class number of Q(3∆).
∆ 23 26 29 31 38 53 59 61 83 87 89 106 107
h3 3 3 3 3 3 3 3 3 3 3 3 3
h01 1 1 1 1 1 1 1 1 1 1 1 3
∆ 109 110 118 129139 157 170 174182 186 199
h3 3 3 3 3 3 3 3 3 3 9
h01 1 1 1 1 1 1 1 1 1 1
We see that for ∆ = 23, 26, 29, 31, 38 etc., K0/K is cyclic of degree 3 and
Z3-embeddable. We shall deal with the remaining cases ∆ = 107,129,174 in the
next section.
Below is the equivalent table for p= 5. Here His the 5-part of the class
number of K(ζ) (with ζa primitive fifth root of unity):
∆ 47 74 79 86 103 119 122 127 131 143 159 166 179 181 194 197
h5555555555555555
H5 5 5 5 5 5 5 25 5 5 5 125 5 5 5 5
We see similarly that the 5-class field K0/K is cyclic of degree 5 and Z5-embeddable
for ∆ = 47, 74, 79 etc. We return to the remaining cases ∆ = 127, 166 in the
next section.
Finally a table for p= 7:
∆ 71 101 134 149 151 173
h7 7 7 7 7 7
H7 7 73737 7
2.3 Algorithm to determine Zp-embeddability
We shall now see how the cases where Theorem 10 does not apply can be dealt
with. Recall that the ray group modulo peis the subgroup Peof Pgenerated
by the principal ideals (α) with integral α1 (mod pe). The ray class group
modulo peis the quotient I/Pe. It is a central result in class field theory that there
is an isomorphism, the Artin symbol, from the p-part of I/Peto Gal(Ke/K).
It maps the p-part of P/Peonto Gal(Ke/K0).
Lemma 14. (I) With notation as above, we have for p > 3,
Gal(Ke/K0)
=(Z/pe1×Z/pe1if p-,
Z/pe1×Z/peif p|.
26
Taking the inverse limit gives Gal(K/K0)
=Zp×Zp. In particular, T= 0 if
K0=K.
(II) For p= 3, the above remains valid when 6≡ 3 (mod 9). Assume
3 (mod 9) and 6= 3. Then
Gal(Ke/K0)
=Z/3e1×Z/3e1×Z/3.
Taking the inverse limit gives Gal(K/K0)
=Z3×Z3×Z/3. In particular,
T
=Z/3if K0=K.
Proof. There is a natural exact sequence
1→ O(O/pe)P/Pe1.
The exclusion of the case p= ∆ = 3 ensures that Ohas trivial p-part. Hence
Gal(Ke/K0) is isomorphic to the p-part of (O/pe)by the Artin symbol. So we
compute the structure of (O/pe).
To begin with, note that each coset of O/pehas a unique representative of
the form a+b∆ with a, b = 0,1, . . . , pe1.
The order of (O/pe), i.e. the norm of the ideal peO, depends on the prime
ideal decomposition of pin K. More precisely, the order of the p-part is
|p-part of (O/pe)|=(p2e2if p-∆ ,
p2e1if p|∆ .
We note the following two facts:
() Let x∈ O and write (1 + x)p= 1 + x0. If pi||xfor some i=1 (meaning
that pi|x, but pi+1 -x), then pi+1||x0.
(∗∗) Let aand bbe integers with a1 (mod p) and pi||bfor some i=1. Write
(a+b∆)p=a0+b0∆. Then a01 (mod p) and pi+1||b0.
It follows from () that the cyclic subgroups U:= h1 + piand V:= h1 + pi
of (O/pe)both have order pe1. It follows from (∗∗) that they have trivial
intersection. So for p-∆,
(p-part of (O/pe)) = U×V
=Z/pe1×Z/pe1.
Assume p|∆. Then U×Vhas index pin the p-part of (O/pe). If p > 3, or
p= 3 and ∆ 6≡ 3 (mod 9), the same argument shows
(p-part of (O/pe)) = U×V0
=Z/pe1×Z/pe
27
for V0:= h1 + i. In case p= 3 and ∆ 3 (mod 9), the 3-part of (O/9)is
h4i×h1+3i×h1 + i
=(Z/3)3, and therefore
(3-part of (O/3e)) = U×V×(group of order 3)
=Z/pe1×Z/pe1×Z/3.
This finishes the proof of the lemma.
The question of Zp-embeddability in the case where the p-class field is cyclic of
degree pcan now be answered.
Theorem 15. Assume K0/K is cyclic of degree p. Pick a prime q-pof Kof
order pin the class group, and write qp= (α)with α∈ O.
I. Suppose p > 3.
(a) If αis not a p’th power in (O/p2), then K0/K is Zp-embeddable (in fact
K0is contained in Kanti), and T= 0.
(b) If αis a p’th power in (O/p2), then K0/K is not embeddable in Z/p2-
extension unramified outside p, and T
=Z/p.
II. Now suppose p= 3. If 6≡ 3 (mod 9), all the above remains valid. Assume
3 (mod 9), and write αa+b∆ (mod 9) with a, b Z.
(c) If (a, b)(±1,0) modulo 3, but not modulo 9, then K0/K is Z3-
embeddable (in fact K0is contained in Kanti), and T
=Z/3.
(d) If (a, b)6≡ (±1,0) modulo 3, then K0/K is embeddable in a Z/9-extension
unramified outside 3, but not in a Z/27-extension unramified outside 3,
and T
=Z/9.
(e) If (a, b)(±1,0) modulo 9, then K0/K is not embeddable in a Z/9-
extension unramified outside 3, and T
=Z/3×Z/3.
Proof. I. Assume p > 3. By Lemma 14, Gal(K/K0)
=Zp×Zp. Since
Gal(K0/K)
=Z/p, there are two possibilities for T: 0 or Z/p.
(a) If T= 0, i.e. Gal(K/K)
=Zp×Zp, then K0is contained in KcyclKanti.
Since K0is dihedral over Q, it is in fact contained in Kanti. Both I/P2and P /P2
have p-rank 2. Therefore, qp= (α) is not ap’th power in P/P2. So αis not a
p’th power in (O/p2O).
(b) If T
=Z/p, i.e. Gal(K/K)
=Zp×Zp×Z/p, then K0is linearly disjoint
from KcyclKanti. So we may for KTtake K0. Hence no Z/p2-extension of Kinside
Kcontains K0. Now the p-part of P/P2is a direct summand in the p-part of
I/P2. Therefore, qp= (α)is ap’th power in P/P2. So αis a p’th power in
(O/p2O).
II. Now assume p= 3. If ∆ 6≡ 3 (mod 9), everything goes like above. Henceforth
28
assume ∆ 3 (mod 9). Then (O/9)
=Z/2×Z/3×Z/3×Z/3 (see the proof
of Lemma 14), so that α=a+b∆ is a cube in (O/9)iff (a, b)(±1,0)
modulo 9. Further, (O/3)
=Z/2×Z/3, so that αis a cube in (O/3)iff
(a, b)(±1,0) modulo 3. By Lemma 14, Gal(K/K0)
=Z3×Z3×Z/3. Since
Gal(K0/K)
=Z/3, there are three possibilities for T:Z/3, Z/9, or Z/3×Z/3.
(c) If T
=Z/3, i.e. Gal(K/K)
=Z3×Z3×Z/3, then K0is contained
in KcyclKanti and therefore also in Kanti. Both I/P2and P/P2have 3-rank 3.
Therefore, q3= (α) is not a cube in P/P2. So αis not a cube in (O/9). On
the other hand, the 3-part of P/P1is a direct summand in the 3-part of I/P1.
Therefore, q3= (α)is a cube in P/P1. So αis a cube in (O/3). This shows the
claims about aand b.
The cases (d) where T
=Z/9 and (e) where T
=Z/3×Z/3 are treated in a
similar manner, so their proofs are omitted.
The same arguments give a description of the torsion subgroup Tin the general
case where K0/K is not necessarily cyclic: If p > 3, or p= 3 and ∆ 6≡ 3 (mod 9),
then K=KcyclKantiK0, and therefore T
=Gal(K0/K0Kanti), i.e. Tis
isomorphic to a subgroup of Gal(K0/K) with cyclic quotient. If p= 3 and
3 (mod 9), then Khas degree 3 over KcyclKantiK0, and therefore Thas a
subgroup of index 3 which is isomorphic to Gal(K0/K0Kanti).
The following examples answer the questions regarding Zp-embeddability from
the previous section and show that all cases of Theorem 15 occur.
Examples. (i) Let p= 5 and ∆ = 127. The class number of Kis 5. The prime
number 2 is divisible by a non-principal prime ideal qof K. Further, q5= (α)
with α= (1 + 127)/2 since 25=α¯α. Since αis not a fifth power in (O/25),
we are in case (a).
(ii) Let p= 5 and ∆ = 166. The class number of Kis 10. Here 7 is
divisible by a non-principal prime ideal qsuch that q5= (α) is principal, α=
(129 + 166)/2. Modulo 25 we have αα5and conclude that we are in case
(b).
(iii) Let p= 3 and ∆ = 107. The class number of Kis 3. Here 11 is divisible by
a non-principal prime ideal qsuch that q3= (α) is principal, α= (9+7