ThesisPDF Available

Procyclic Galois Extensions of Algebraic Number Fields



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
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
2006 David Brink (according to the Danish legislation)
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
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
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.
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
(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
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
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).
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
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
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
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 )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
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
If Xhas finite rank, the Elementary Divisor Theorem gives
where a < is the essential rank of X, and Tis a finite p-group.
Let us finally note that if
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
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.
For sufficiently large n, the p-adic logarithm is an isomorphism
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
is a Zp-module of rank
[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-
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
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.
1.5 The maximal number of linearly disjoint Zp-
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-
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
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.
Up, U00 =Y
Up, U =U0×U00.
Uis an open subgroup of J. The normgroup of the maximal abelian extension
of Kunramified outside pis
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
and H0has finite index in J(because J/H0is isomorphic to K’s class group).
Evidently U0H=H0, so
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
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)
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
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.
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
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
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
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
(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\
Evidently one has V=Vand the inclusions
The quotient VS/Kpis a vectorspace over Fp, the dimension of which is denoted
σ(S). First we compute σ():
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
V C, 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:
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
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
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
with US=Qp6∈SUp. So we have to compute the dimension of
Consider the following sequence of vectorspaces over Fp:
1 VS/Kpf4
J/N 1.
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
p,dim(U/USUp) = X
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
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
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(∆)
ior p
iis a square in Q2
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
In this case Kis called p-rational. This notion was introduced in [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
=Gal(L/K)o Z/2.
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)
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], Theor