1. Ramification Theory
Guillem Sala Fernández
Universitat de Barcelona
Abstract.
The aim for this paper is to provide an adequate language in
order to describe ramification in not necessarily finite Galois
extensions.
1 Introduction
Our goal is to give a first approach to the study of infinite algebraic
extensions and understand the properties of ramification in such ex-
tensions. In order to do so, we will give some background tools that
may be needed in order to follow the proofs along this paper in the
preliminaries. The structure of this paper will be the following.
Firstly, let A be an integrally closed domain of Krull dimension 1, K
its fraction field, L|K a finite or infinite Galois extension and B the
integral closure of A in L. We will show that B is also 1-dimensional
integrally closed domain, and also that if A is a Dedekind ring, B
is not, in general, a Dedekind domain, in comparison with the finite
case, in which it has been previously seen that it is.
Secondly, let us consider P ⊂ B a non-zero prime ideal of B and p :=
P ∩ A its contraction to A. We will give a general definition of the
inertia and decomposition groups, that we will denote as G0(P, p)
and G−1(P, p). Following its definition, we will give some results that
show its application to ramification theory.
In fact, we will see that if L can be written as a countable union of
fields, say Kn, of finite degree and Galois of K, then Gal(L|K) acts
transitively on the set of prime ideals P of B such that P ∩ A = p,
the residual extension B/P|A/p is normal, the decomposition and

2. 2 Guillem Sala Fernández
inertia groups are closed subgroups of Gal(L|K) and there exists an
exact sequence of morphisms between topological groups, namely
1 → G0(P|p) → G−1(P|p) → Gal(B/P|A/p) → 1.
After that, we will give a precise definition of non-ramification and
total ramification of prime ideals of B over their contractions in A.
To conclude, given a finite set of prime numbers, S, we will give a
description of the maximal non-ramified abelian extension outside
S, which we will denote as QS
ab|Q, the structure of its Galois group
and, in addition, study its decomposition and inertia groups.
Preliminaries
Here are some of the results and definitions that we will need in order
to develop this theory. Some of them can be given in a more general
way, but we will give the version that better suits us according to
our needs. Let us start by defining
Definition 1 Let A be a partially ordered index set and let {Ga}a∈A
be a family of groups. Assume that:
1. For all a, b ∈ A with a ≤ b there exists a group homomorphism
φb,a : Gb → Ga, such that if a ≤ b ≤ c, then φb,a ◦ φc,b = φc,a.
2. For all a ∈ A, φa,a = idGa .
We will say that {Ga, φb,a} is a projective family of groups.
For instance, suppose m ≤ n if and only if m|n. Then, under this
partial order, we can take the family of groups {Z/nZ}n∈N together
with the reduction morphisms, φn,m : Z/nZ → Z/mZ and φn,n :
Z/nZ → Z/nZ given by the identity, and it is easy to see that
{Z/nZ, φm,n} is a projective family. This example will be of great
use when we give a definition of the p-adic integers.
Definition 2 We say that a group G is a projective limit of the
projective family {Ga, φb,a} if it is a group together with a family of
homomorphisms φa : G → Ga such that the following conditions hold
1. If a ≤ b, then φb,a ◦ φb = φa and

3. Ramification Theory 3
2. Given any group H together with a family of homomorphisms
ρa : H → Ga satisfying the previous condition, there exists a
unique group homomorphism χ : H → G such that φa ◦ χ = ρa
for all a ∈ A.
We will denote the projective limit as G = lim
←−
Ga.
It is known that for any projective family of groups, there exists a
projective limit and that if two groups G and H are projective limits
of a projective family, then the group homomorphism χ : H → G of
the condition 2 in definition 2 is a group isomorphism.
On the other hand, there is another notion that we should remember,
the p-adic integers, which will be mentioned by the end of the final
section, non-finite abelian extensions of Q. Since it can be shown
that
lim
←−
a∈A
Ga = (ga)a ∈
a∈A
Ga : gc = φb,c(gb), ∀c ≤ b ∈ A ,
in order to do define the p-adic integers, all we need to do is consider
the projective family {Z/pk
Z, φl,k}k where φl,k denotes the natural
projection for k ≤ l,
φl,k : Z/pl
Z −→ Z/pk
Z.
Definition 3 We will define the set of p-adic integers, Zp, as the
projective limit of the projective family {Z/pk
Z, φl,k}k, that is
Zp = lim
←−
k∈N
Z/pk
Z = (xl)l ∈
∞
l=1
Z/pl
Z : xk = φk ,k(xk ), ∀k ≤ k ∈ N
This leads to another example that will be treated in another lecture:
the profinite completion of Z. Taking the projective family from the
first example, the completion of Z is given by
ˆZ := lim
←−
n∈N
Z/nZ
p prime
Zp
Definition 4 A group G is a profinite group if G = lim
←−
Ga for a
projective family {Ga, φb,a}, where each Ga is a finite group.

4. 4 Guillem Sala Fernández
In fact, a natural topology can be built on any given profinite group.
That is, let G = lim
←−
Ga be a profinite group. We can endow each Ga
with the discrete topology, then give a∈A Ga the product topology
and finally give lim
←−
Ga ⊂ a∈A Ga the subspace topology. The previ-
ous inclusion is due to the fact that the projective limit is a subset of
a∈A Ga, as we saw right before the definition of the p-adic integers.
Furthermore, there is a result that shows that, under the previous
topology, G is a topological group.
On the other hand, there also is a result that shows that if G is
a profinite group equipped with the topology described above then
G, as a topological group, is compact, Hausdorff and totally discon-
nected.
Now we have all the tools we need in order to endow the Galois
group of an algebraic extension with a topology, but in order to do
so, we need to define infinite algebraic extensions of fields.
Proposition 1 A field extension of infinite degree L|K is algebraic
if L is the union of all K with K ⊂ L such that K |K is a finite
algebraic extension. Also, we will say that L|K is a Galois extension
if it is normal and separable.
In addition to this, we can define the Krull topology of a Galois
extension, but first notice that from now on we will refer to Galois
extensions regardless of its order, that is, whether they are finite or
not.
Definition 5 Let L|K be a Galois extension, and let G = Gal(L|K).
The Krull topology on G is the topology with basis all cosets σH,
where σ ∈ G, H = Gal(L|K ), and K |K is a finite Galois extension.
Actually, acording to the next proposition, it can be shown that this
basis generates a topology that is equal to the profinite topology
that we saw before.
Let us denote E(L|K) as the set of all subextensions of L|K of the
form K |K, Ef (L|K) as the set of all finite Galois subextensions of
L|K, S(L|K) as the set of all subgroups of Gal(L|K) and F(L|K)
as the set of all closed subgroups of Gal(L|K).

5. Ramification Theory 5
Proposition 2 With the previous notation, if we endow G with
the Krull topology and let lim
←−K ∈Ef (L|K)
Gal(K |K) be the projec-
tive limit of the projective family {Gal(K |K), red}K ∈Ef (L|K) together
with the profinite group topology then, Gal(L|K) homeomorphic to
lim
←−K ∈Ef (L|K):
Gal(K |K). Also,
1. The Galois group G forms a topological group.
2. The Galois group G is compact, Hausdorff and totally discon-
nected.
So we are finally ready to give the Fundamental Theorem of Infinite
Galois Theory
Theorem 1 Let L|K be a Galois extension, and let G = Gal(L|K)
be its Galois group. Then,
1. with the Krull topology on G the maps
E(L|K) −→ F (S(L|K))
K |K −→ Gal(L|K)
and
F (S(L|K)) −→ E(L|K)
H −→ LH
|K
give an inclusion-reversing correspondence between F (S(L|K)),
and the set of intermediate fields of L|K, E(L|K).
2. If K corresponds to H, then the following are equivalent:
(a) [G : H] < ∞.
(b) [K : K] < ∞.
(c) H is open.
3. If any of the equivalent conditions in 2. is satisfied, |G : H| =
[K : K].
4. For any closed subgroup H ≤ G, where H = Gal(L|K ), we have
H G if and only if K |K a normal extension. If this is the case
then there exists a group isomorphism θ : G/H → Gal(K |K).
As we can see, if L|K is a Galois extension with [L : K] < ∞, then
the Krull topology on Gal(L|K) is discrete, due to the fact that
L|K is a finite Galois extension and, therefore, Gal(L|L) = {id} is
an open set. From this follows that every subgroup of Gal(L|K) is
closed and the previous correspondence becomes a bijection.

6. 6 Guillem Sala Fernández
2 Ramification in Infinite Extensions
Let A be a 1-dimensional integrally closed domain with fraction field
K := K(A), let L|K be a Galois extension and let B be the integral
closure of A in L. The following proposition shows that B inherits
the properties of A.
Proposition 3 With the previous hypothesis, B is a 1-dimensional,
integrally closed domain.
Proof. Since B is the integral closure of A in L, B|A is an integral
extension and B is an integrally closed domain. Also, since A is a
1-dimensional ring and, as we have seen before, B|A is an integral
extension, B is a one dimensional ring.
The main difference with respect to finite Galois extensions is that if
L|K is an infinite Galois extension, it is not true in general that if A
is a Dedekind Domain, then B is too. To show this fact, we will give
an example of an extension in which A has the property of being a
Dedekind domain but B does not have it.
Let p be a prime number and, for each integer n ≥ 1, consider a
primitive pn
-th root of unity, say ζn, the field sequence Kn := Q(ζn),
and
K∞ :=
∞
n=1
Kn.
Acording to Chapter 3 of the course notes, the ring of integers of
each of the fields Kn is On := Z[ζn] (cf. Thm 3.9.9). Also, the integral
closure of K∞ is
O∞ :=
∞
n=1
On.
On the other hand, if we let Pn be the ideal of On generated by
1 − ζn, we have that Pn is the unique prime ideal of On that divides
p. That is due to the fact that, just like we saw in the course notes
(cf. 3.9.2), for each n ∈ N, (1 − ζn)On is a prime ideal and
pOn = ((1 − ζn)On)ϕ(pn)
= ((1 − ζn)On)pn−1(p−1)
.

7. Ramification Theory 7
Now, the ideal P∞ of O∞, generated by all the elements 1−ζn, is in
fact P∞ = ∪∞
n=1Pn. Hence, P∞ is not the total ideal of O∞ and it
is also a maximal ideal with residual field Fp, which is actually the
residual field of Pn in On for each n ∈ N. However, the prime ideal
P∞ is exactly equal to its own p-th power, since the ideal (1−ζn)On
is the p-th power of the ideal (1 − ζn+1)On+1. Therefore,
Pp
∞ =
∞
n=1
((1 − ζn)On)p
=
∞
n=1
(1 − ζn−1)On−1 =
∞
j=1
(1 − ζj)Oj = P∞
Consequently, O∞ doesn’t admit a unique factorization in prime
ideals and thus, O∞ cannot be a Dedekind domain, otherwise it
would contradict Theorem 2.4.5 from the course notes, namely, it
would contradict Dedekind’s unique factorization theorem.
Despite the fact that B doesn’t inherit the property of A of being a
Dedekind domain, we can still define the decomposition and inertia
groups of a non-zero prime ideal P ⊂ B over its contraction to A,
p := P ∩ A, just as follows.
From now on, we will suppose that L can be written as a countable
union of fields, which we will denote as Kn, which are finite Galois
extensions of K. We will also assume that Kn ⊂ Kn+1, that is,
{Kn}n∈N is an increasing sequence of fields.
Notice that this last assumption doesn’t affect the generality of the
results that we are about to give, since we will be working on count-
able fields, for instance, any not necessarily finite algebraic number
field over the rational numbers Q works just fine.
Now we will enunciate and prove, in a more general way, a result
that has previously been shown in the finite case.
Proposition 4 Let p be any prime ideal of the ring A. Then, the
Galois group Gal(L|K) acts transitively on the set of prime ideals
P ⊂ B such that P ∩ A = p.
Proof. Let P, P ⊂ B be two prime ideals of B such that P ∩ A =
p = P ∩ A. Let {Kj}∞
j=0 a sequence of Galois subextensions such
that
K =: K0 ⊂ K1 ⊂ · · · ⊂ Kj ⊂ Kj+1 ⊂ · · · ,

8. 8 Guillem Sala Fernández
and such that L = ∪∞
j=0Kj. Also, let us denote by An the integral
closure of A in Kn and let Pn := P ∩ An and Pn := P ∩ An.
Since Kn|K is a finite Galois extension, there exists σn ∈ Gal(Kn|K)
such that σn(Pn) = Pn, because the result we are trying to prove is
true in the case of finite extensions (cf. Prop. 3.5.1). Even though in
Prop. 3.5.1 of the course notes A is a Dedekind domain, the result
is still true under the sole hypothesis that A is integrally closed,
because the Noetherian and 1-dimensional properties are not used
anywhere in the proof.
We aim to prove that the sequence (σn)∞
n=0 verifies that σn+1|Kn = σn
for all n ≥ 0. If we get that equality, we will have that there exists
σ ∈ Gal(L|K) such that the restriction of σ to each of the fields
Kn is eventually σn for each n ≥ 0, because just like we saw in the
preliminaries,
Gal(L|K) lim
←−
Gal(Kn|K),
thus, the prime ideal σ(P) of B is actually
σ(P) =
n≥0
σ(Pn) =
n≥0
Pn = P .
In such manner, let us choose the right automorphisms σn so that
the restriction of σn+1 to the field Kn is exactly σn and σn+1(Pn+1) =
Pn+1.
In order to do it, let τn+1 ∈ Gal(Kn+1|K) be any extension of σn to
the field Kn+1. Then, τn+1(Pn+1) is a prime ideal of An+1, because
the Galois group acts transitively in finite extensions. Besides, since
τn+1(Pn+1) and Pn+1 are prime ideals of An+1 such that τn+1(Pn+1)∩
An = Pn = Pn+1 ∩ An, which is a prime ideal of An and Kn+1|Kn
is a finite Galois extension, there exists ρn+1 ∈ Gal(Kn+1|Kn) such
that ρn+1(τn+1(Pn+1)) = Pn+1. we can set σn+1 := ρn+1 or τn+1 ∈
Gal(Kn+1|Kn).
Finally, σn+1 extends σn because we are assuming τn+1 is an exten-
sion of σn and ρn+1 acts as the identity in Kn, apart from sending
Pn+1 to Pn+1.
And finally, we can define both decomposition and inertia groups.

9. Ramification Theory 9
Definition 6 Given the previous notations, let G denote the Galois
group of L|K, G = Gal(L|K). We define the decomposition group of
P over p as
G−1(P|p) := {σ ∈ G : σ(P) = P} ,
and the inertia group of P over p as
G0(P|p) := {σ ∈ G−1(P|p) : σ(b) ≡ b mod P, ∀b ∈ B}
This definition can be applied to the case in which A is the algebraic
closure of Z in a not necessarily finite algebraic extension of fields
of Q, and also in a symmetrized ring of A with respect to any mul-
tiplicatively closed system. These two cases are the ones that truly
concern us in this paper.
Now, under the same assumptions, that is, considering that L can
be written as a union of an increasing sequence of Galois extensions,
Proposition 5 Let P ⊂ B be a non zero prime ideal of B and
p := P ∩ A its contraction to A. Then, the following properties hold.
1. The residual extension (B/P)|(A/p) is a normal algebraic exten-
sion of fields.
2. The decomposition and inertia groups, G−1(P|p) and G0(P|p),
are closed subgroups of Gal(L|K).
3. The natural sequence of topological group morphisms,
1 → G0(P|p) → G−1(P|p) → Gal(B/P|A/p) → 1,
is an exact sequence.
Proof. Just like in the last proof, let us fix An := B ∩ Kn and Pn :=
P ∩ An, for each n ≥ 0. Then,
B =
n≥0
An, P =
n≥0
Pn, and B/P =
n≥0
(An/Pn) .
Consequently, (B/P)|(A/p) is a normal algebraic extension, because
each (An/Pn) |(A/p) is, in effect, a normal algebraic extension, and

10. 10 Guillem Sala Fernández
the decomposition and inertia groups, G−1(P|p) and G−1(P|p) sat-
isfy that
G−1(P|p) lim
←−
n∈N
G−1(Pn|p) and G0(P|p) lim
←−
n∈N
G−1(Pn|p),
hence, both are closed subgroups of Gal(L|K). On the other hand, we
have previously seen (cf. Prop. 3.5.2) that there exists a commutative
diagram with exact rows, namely,
1 G0(Pn+1|p) G−1(Pn+1|p) Gal An+1
Pn+1
|A
p
1
1 G0(Pn|p) G−1(Pn|p) Gal An
Pn
|A
p
1,
res res res
where the vertical morphisms are given by the restriction to each of
the subgroups and for the horizontal exactitude, all we need to take
is the injection i, and π from Prop. 3.5.2, which is exhaustive and
also induces the isomorphism Coker(i) := G−1(Pn|p)/G0(Pn|p)
Gal An
Pn
A
p
thanks to that same proposition. Therefore, the se-
quence of morphisms
1 −→ G0(Pn|p)
i
−→ G−1(Pn|p)
π
−→ Gal
An
Pn
A
p
−→ 1,
is exact. Therefore, through using the projective limit and the pre-
vious exact sequence we get the following sequence of morphisms,
which is actually an exact sequence of profinite groups.
lim
←−n∈N
G−1(Pn|p) lim
←−n∈N
G−1(Pn|p) lim
←−n∈N
Gal An
Pn
|A
p
1 G0(Pn|p) G−1(Pn|p) Gal An
Pn
|A
p
1.

11. Ramification Theory 11
With this last proposition we can give a precise definition of non-
ramification and total ramification of prime ideals.
Definition 7 We say that a prime ideal P is non-ramified over its
contraction p when the inertia group G0(P|p) is a trivial group, that
is, when all the ideals Pn are non-ramified over p. In the same way,
we say that P is totally ramified when the inertia group is equal to
Gal(L|K), namely, all the prime ideals Pn are totally ramified over
p.
Now we can proceed to the final part of this paper.
3 Non-finite abelian extensions of Q
During all this section, we will let K = Q be the base field. We will
make a general study of the ramification properties of the abelian
extensions of Q. Also we will let p be a prime number. Let’s give a
first definition:
Definition 8 We will denote µ(p∞
) as the group of all roots of unity
of order a power of p in a fixed algebraic closure of Q, that is
µ(p∞
) := {ζ ∈ C : ζpr
= 1, for some r ∈ Z}.
Actually, we can define µ(pk
) := {ζ ∈ C : ζpk
= 1} and therefore
µ(p∞
) =
∞
k=1
µ(pk
).
Given that definition, we observe that Q(µ(p∞
))|Q is an abelian
extension, because both Q(µ(pk
))|Q and Q(ζpk )|Q are equal and
Q(µ(p∞
)) =
∞
k=1
Q(µ(pk
)).
Also, its Galois group, Gal(Q(µ(p∞
))|Q), is isomorphic to
Z/(p − 1)Z × lim
←−
Z/pn
Z Z/(p − 1)Z × Zp, p = 2,
Z/2Z × lim
←−
Z/2n
Z Z/2Z × Z2, p = 2,

12. 12 Guillem Sala Fernández
because of the bijection given by the pk
-th cyclotomic character,
χpk : Gal(Q(ζpk )|Q) → (Z/pk
Z)∗
. Namely,
Gal(Q(ζpk )|Q)
χpk
(Z/pk
Z)∗
Z/pk−1
(p − 1)Z
CRT
Z/pk−1
Z × Z/(p − 1)Z,
and if we take projective limits, then
Gal(Q(µ(p∞
))|Q) lim
←−
n∈N
Gal(Q(ζpn )|Q)
lim
←−
n∈N
Z/pn−1
Z × Z/(p − 1)Z
(∗)
Z/(p − 1)Z × lim
←−
n∈N
Z/pn−1
Z Z/(p − 1)Z × Zp,
where (∗) is due to the definition of projective limit via coherent se-
quences. The case where p = 2 can be seen by following the previous
steps.
In addition to this, Q(µ(p∞
))|Q is a totally ramified extension in p
and non-ramified outside p. Furthermore, it is the maximal abelian
extension field of Q such that it is non-ramified outside of p. Actually,
we can give a more general description of this fact. The proof follows
from the previous observation.
Proposition 6 Let S denote any set of prime numbers. We will
denote QS
ab|Q as the maximal abelian extension of Q such that it is
non-ramified in any l /∈ S. Then, the extension QS
ab|Q verifies the
following properties:
1. QS
ab p∈S Q(µ∞(p)).
2. Its Galois group is isomorphic to the product of the Galois groups
of each of the extensions Q(µ∞(p))|Q for each p ∈ S, namely
Gal(QS
ab|Q) p∈S Z/(p − 1)Z × p∈S Zp, if 2 /∈ S,
Z/2Z × p∈S Z/(p − 1)Z × p∈S Zp, if 2 ∈ S.

13. Ramification Theory 13
On the other hand if S is a finite set, then the Galois group of QS
ab|Q
is a finitely generated topological group, due to the fact that the
product of the groups Zp for each p ∈ S is procyclic, that is, it is
a projective limit of cyclic groups, and therefore it is topologically
generated by a unique element. As a consequence, the number of
topological generators of Gal(QS
ab|Q) is ≤ #(S) + 1.