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Valuations, Local Fields, and Teichmüller Representatives
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U NIVERSITY OF WATERLOO D EPARTMENT OF P URE M ATHEMATICS
Valuations, Local Fields, and Teichmüller
Representatives
Nicolas Banks
April 14, 2021
1 I NTRODUCTION
Class field theory is, broadly speaking, the study of Abelian extensions of Q - that is, extensions of Q whose
automorphism groups are Abelian. We have seen in our course that cyclotomic fields are examples of Abelian
number fields. The remarkable Kronecker-Weber Theorem asserts that, in fact, these are the only examples. This is
a crowning achievement of class field theory, and its many important consequences include a vast generalization of
the law of quadratic reciprocity.
Class field theory is a huge subject with connections to many areas of mathematics, including group cohomology
and infinite Galois theory. To this end, we present an introduction to the theory of local fields; the study of extensions
of such fields leads to local class field theory, from which the "global" theory can be built.
A defining trait of class field theory is the use of topological methods. This arises naturally from the study of
infinite extensions and from the employment of valuations and absolute values. Therefore, we briefly introduce
some topological notions that support this theory.
This will all culminate in the definition of the Teichmüller map for any local field. In a much more general setting,
this map is used to study deformations of Galois representations. This theory was notably used in the proof of
Fermat’s Last Theorem, published in 1994, whereby Andrew Wiles used Galois deformations to show that a solution
to Fermat’s equation leads to the existence of an elliptic curve which cannot exist.
Unless otherwise indicated, the main reference for this report is Kenichi Iwasawa’s Local Class Field Theory, [3].
1
2 VALUATIONS ON F IELDS
Throughout, let k be a field. We will make use of the extended real numbers R := R ∪ {∞}, where ∞ is a formal
symbol satisfying various properties, c.f. [7]. Notably, we do not define 0 · ∞, ∞ − ∞, or ∞/∞.
Definition 2.1. A valuation on k is a map
v :k →R
such that, for all x, y ∈ k,
• v(x) = ∞ if and only if x = 0;
• v(x + y) ≥ min(v(x), v(y)); and
• v(x y) = v(x) + v(y).
Remark 1. We will not have occasion to make use of this, but the above can be made more general by replacing R
with any totally ordered Abelian group Γ. One defines Γ := Γ ∪ {∞} in a similar fashion as R. Interested readers are
directed to [B].
k can always be endowed with the trivial valuation v 0 , defined as v 0 (0) = ∞ and v 0 (x) = 0 for all x ∈ k ∗ . The
following gives a class of more interesting examples.
Example 2.1. Fix a prime number p ∈ Z. For any x ∈ Q∗ , we can uniquely express x = p a m
n , where m and n are
coprime to p. We think of a as the difference of the highest power of p dividing the numerator of x and that of the
denominator of x. Clearly a ∈ Z. Thus we can define the p-adic valuation
v p : Q → Z ∪ {∞}
³ m´
v p (x) = v p p a
= a, x ∈ Q∗ , and v p (0) = ∞.
n
Observe that v p (p) = 1.
Example 2.2. Consider the field k(t ) of rational functions in the indeterminate t with coefficients in k. Since k(t )
is the fraction field of the polynomial ring k[t ], which is a UFD, we can construct a similar class of valuations on
p
k(t ). Specifically, if f (t ) ∈ k[t ] is irreducible, then we can express any r (t ) ∈ k(t )∗ uniquely in the form r = f a q , where
a ∈ Z and p, q are coprime to f . Then we define v f (0) = ∞ and v f (x) = a for all x ∈ k(t )∗ when expressed in the
aforementioned form. We note that v f satisfies v f ( f ) = 1. This valuation is sometimes called the f -adic valuation.
Proposition 2.1. Let v be a valuation on the field k. Then the following hold:
• v(1) = v(−1) = 0;
• v(x) = v(−x); and
• if v(x) < v(y), then v(x + y) = v(x).
Proof. Immediately from the definition.
There are three important objects associated with a valuation.
2
Definition 2.2. Let v be a valuation on k.
• The subset O v := {x ∈ k | v(x) ≥ 0} is called the valuation ring of v.
• mv := {x ∈ k | v(x) > 0} is called the maximal ideal of v.
• F v := O v /mv is the residue field of v.
We immediately justify the terminology with the following theorem.
Theorem 2.1. Let v be a valuation the field k. Then O v is a subring of k, in fact an integral domain, and mv is a
maximal ideal of O v .
Proof. By definition of ∞, it is clear that 0 ∈ O v and 0 ∈ mv . Further, the previous proposition shows that 1 ∈ O v .
Now let x, y ∈ O v , so that v(x), v(y) ≥ 0, and without loss of generality take v(y) ≥ v(x). Then since v is a valuation,
we have
v(x + y) ≥ v(x) ≥ 0,
so x + y ∈ O v . Similarly, we have
v(x y) = v(x) + v(y) ≥ 0 + 0 = 0,
so x y ∈ O v , and O v is a subring of k. The fact that O v is an integral domain is obvious from the fact that it is contained
in the field k.
Since v(1) = 0 and 0 ∈ mv , mv is a proper, non-empty subset of O v . A virtually identical proof to the above shows
that mv is an ideal of O v . To see that mv is maximal, we will prove that the units of O v are precisely the elements
with valuation 0. This will show that mv consists precisely of the non-units of O v , proving maximality.
To this end, let x ∈ O v \ {0}. Then x clearly has an inverse x −1 in k ∗ , and we merely need to check when x −1 ∈ O v .
Observe that
¡
¢
¡
¢
v xx −1 = v(1) = 0 = v(x) + v x −1
¡
¢
⇒ v x −1 = −v(x),
so x −1 ∈ O v if and only if v(x) = 0, as required.
Remark 2. The above proof shows that mv is actually the unique maximal ideal of O v , as any other maximal ideal
would necessarily contain a unit and thus be the entire ring O v , contradicting maximality. Thus O v is a local ring.
We will not deeply explore the theory of local rings here, but they are of fundamental importance in algebraic
geometry and commutative algebra. See for example [1].
Example 2.3. Equip the rational numbers Q with the p-adic valuation v p for some fixed prime p ∈ Z. Then it is not
hard to see that
O v p = Z(p) =
na
b
o
| a, b ∈ Z, p - b ,
which is precisely the localization of Z at the ideal ⟨p⟩ = pZ. We also see that mv = pZ(p) and F v = Fp , the finite field
with p elements.
Changing to the case of an f -adic valuation on the function field k(t ), we obtain completely analogous results to
the above.
Definition 2.3. Let v be a valuation on k. The unit group of v, denoted U v , is the multiplicative group O v∗ of units
in the valuation ring of v. Note that the proof of Theorem 2.1 shows that U v = ker(v).
3
By the defining properties of v, we see that
v : k∗ → R
is a group homomorphism from the multiplicative group of non-zero elements of k to the additive group of real
numbers. Then the image v(k ∗ ) is an additive subgroup of R, and by the first isomorphism theorem we obtain a
group isomorphism
∗
k ∗ /U v ∼
= v(k ).
Definition 2.4. The valuation v on k is called discrete if the subgroup v(k ∗ ) of R is a 1-dimensional lattice - in other
words, if v(k ∗ ) is a discrete subgroup of R and hence can be written as
v(k ∗ ) = Zα
for some α ∈ R. If v is discrete with α = 1, then we say that v is normalized. A ring which arises as the valuation ring
of a discrete valuation on some field is called a discrete valuation ring or DVR.
Definition 2.5. Let µ be another valuation on k. We say that v and µ are equivalent, written v ∼ µ, if there exists a
positive real number α such that α · v = µ.
Remark 3. It is trivial to prove that the above forms an equivalence relation on valuations on a given field k.
Moreover, since equivalent valuations are required to be positive multiples of each other, we see that equivalent
valuations yield the same valuation rings, maximal ideals, residue fields, unit groups, and (to be discussed later)
topologies. For all intents and purposes, their properties are the same.
In particular, if v is a discrete valuation then we can always replace v with an equivalent normalized valuation.
Therefore, going forward we will assume that v(k ∗ ) = Z whenever v is discrete.
Example 2.4. The localized integers Z(p) with the p-adic valuation are a DVR for any prime p ∈ Z. In fact, the
localization of any Dedekind domain at a non-zero prime ideal is a DVR. The connection between DVRs and Dedekind
domains is very deep, c.f. [8]. For a non-example of a DVR, see the Hahn series over Q, [2]. In fact, this example is
particularly fruitful, as it forms a complete valuation ring (to be discussed soon) with finite residue field which is
nevertheless not discrete.
Remark 4. The theory of valuations and valuation rings has a dual formulation to what is presented here. In this
paper, we begin with a field and a valuation on that field, and the valuation ring arises as a distinguished subring of
our field. We could have instead started with an integral domain R and taken its field of fractions k. However, not all
integral domains can be equipped with a valuation which has desirable properties. Surprisingly, a necessary and
sufficient condition on R is that for any element x of the field of fractions k, either x or x −1 is in k. One can then
show that R is a local ring and that a valuation can be defined on R, which can be extended in a natural way to k.
This equivalence is not trivial to prove; interested readers are directed to [4].
3 T OPOLOGICAL AND C OMPLETE F IELDS
Throughout, let k be a field and v be a valuation on k.
Definition 3.1. We call k a topological field if it is a topological space and the addition, multiplication, and inversion
maps, defined below, are continuous:
A : k × k → k, (x, y) 7→ x + y
4
M : k × k → k, (x, y) 7→ x · y
ι : k → k, x 7→ x −1 .
Naturally, we endow k × k with the product topology.
Definition 3.2. The v-topology on k is the topology whose base is given by sets of the form
B r,v (x) = {y ∈ k | v(y − x) > r },
where x ∈ k and r ∈ R. Explicitly, a subset of k is open in the v-topology if it can be written as an arbitrary union of
sets of the form B r,v (x). Such sets are called open balls centered at x of radius r .
Remark 5. Henceforth, when referring to a topology on the field k or to a topological property of k, we assume it is
the v-topology for a fixed valuation v unless otherwise stated.
Theorem 3.1. The map
v : k∗ → R
is continuous, where R has the standard topology whose base is given by open intervals.
Proof. Let U ⊆ R be open; we wish to prove that v −1 (U ) is open in k. Since preimages preserve unions, we need
only check that this holds when U is an open interval.
So, let U = (a, b) for a, b ∈ R. Then
v −1 (U ) = {x ∈ k | a < v(x) < b} = {x ∈ k | v(x) > a} ∩ {x ∈ k | v(x) < b}
¡
¢
= {x ∈ k | v(x − 0) > a} ∩ {x ∈ k | v x −1 − 0 > −b}
= B a,0 ∩ B −b,0 .
Since a finite intersection of open sets is open, this proves that v −1 (U ) is open. A similar proof applies when a or b
is infinite.
Theorem 3.2. The v-topology on k turns k into a topological field. Further, O v is closed in k, and mv is open in k.
Proof. The fact that addition, multiplication, and inversion are continuous follows easily from continuity of v. Now
observe that O v = {x ∈ k | v(x) ≥ 0} = v −1 ([0, ∞)) is the preimage of a closed set under a continuous map, hence is
closed. Similarly, mv = v −1 ((0, ∞)) is open.
Definition 3.3. A sequence (x n )∞
n=1 of elements of k converges or is convergent if there exists some x ∈ k such that
lim v(x n − x) = ∞.
n→∞
∞
In this case, we say that (x n )∞
n=1 converges to x and that x is the limit of (x n )n=1 , also written as x n → x.
Remark 6. It is a standard topological result that limits of convergent sequences are unique, justifying the use of the
word "the" in the above definition. Further, continuity of v implies that x n → x if and only if
lim v(x n ) = v(x).
n→∞
5
Another standard topological argument shows that, if x n → x for some x 6= 0, then there exists some N ∈ N such that
v(x n ) = v(x) for all n > N .
Definition 3.4. A sequence (x n )∞
n=1 of elements of k is called Cauchy if v(x m − x n ) → ∞ as m, n → ∞.
Remark 7. Informally, the points of a Cauchy sequence get closer to each other, and the definition does not make
reference to a particular limiting point. Clearly, every convergent sequence in k converges in k, but the converse
need not hold. Typically, the failure to converge happens when the point that "should" be the limit of the Cauchy
sequence ends up lying outside of the field k. Students of analysis will be familiar with this phenomenon from their
study of the rational numbers with the standard metric. This motivates the following class of valuations.
Definition 3.5. The valuation v is called complete if every Cauchy sequence in k converges to a point in k.
Example 3.1. The field Q with any p-adic valuation is not complete, which should be intuitive from the corresponding
fact about the standard absolute value on Q. However, one can always (in an essentially unique way) construct a
completion of a field, as in [H]. With the standard absolute value on Q, this is precisely the construction of R. With
the p-adic valuation, this gives the p-adic numbers Qp . These can be viewed as formal power series in powers of p
with rational coefficients, and the corresponding valuation ring is the p-adic integers Zp . The categorically-minded
reader will recognize Zp as the colimit (also projective limit or inverse limit) of the sequence of rings Z/p i Z, i ∈ N; see
[6] for details.
Example 3.2. Let F be any field. The field k := F ((t )) in the indeterminate t denotes the field of formal Laurent series
∞
X
ai t i ,
i =−m
where m ∈ N and a i ∈ F . We define a valuation v on k by v(0) = ∞, and v( f ) is the index of the first non-zero
coefficient of a non-zero Laurent series f . Then k turns out to be a complete field; in fact, F ((t )) is the completion of
F (t ) as described in Example 2.2.
4 U NIFORMIZERS
Throughout, let k be a field and v a discrete (and, without loss of generality, normalized) valuation on k. Since
v(k ∗ ) = Z, there exists an element π ∈ k ∗ (not necessarily unique) such that v(π) = 1. Such elements, which by
definition lie in the valuation ring of v, describe the algebraic structure of the valuation v in a remarkably simple
way.
Definition 4.1. An element π ∈ k ∗ such that v(π) = 1 is called a uniformizer of k (also variously called a prime
element and a uniformizing parameter of k).
Theorem 4.1. Let π be a uniformizer of k. Then mv = ⟨π⟩ = πO v . Further, every non-zero ideal I of O v can be written
in the form I = mnv = ⟨πn ⟩ for some n ∈ N. In particular, O v is a PID.
Proof. Since v(π) = 1 by definition, it is clear that π ∈ mv , so by minimality of ⟨π⟩, we have ⟨π⟩ ⊆ mv . For the other
inclusion, it is clear that 0 = 0π, so let x ∈ mv \ {0}. Since v(x) > 0 and v is discrete, we can write v(x) = n for some
n ∈ N. Then observe that
x=
x
· πn .
πn
6
x
, we see that v(u) = v(x) − v(πn ) = v(x) − nv(π) = v(x) − n = n − n = 0, so that u ∈ O v∗ . In particular,
πn
we have written x = uπn = uπn−1 π as an O v -multiple of π, proving that mv ⊆ ⟨π⟩.
Setting u =
Now let I be any non-zero ideal of O v . Then there exists an element y ∈ I such that v(y) = n is a non-negative
integer, and moreover, we can take such a y with n minimal. Then a virtually identical proof to the above shows that
I = ⟨πn ⟩, as required.
Corollary 4.1. Any two uniformizers of k are unit multiples of each other.
Proof. By the previous theorem, any two uniformizers are generators of the same principal ideal, namely mv . Since
O v is an integral domain, this implies that the two generators are unit multiples, as required.
Remark 8. This gives an alternative way to view the valuation on O v - namely, as the unique non-negative integer n
such that x = uπn where u is a unit in O v . Since any unit has valuation 0, the above Corollary 4.1 shows that this is
independent of the choice of uniformizer.
Example 4.1. The prime number p is a uniformizer of Q with the p-adic valuation.
5 T HE T EICHMÜLLER M AP
Local fields are at the heart of local class field theory, as the name implies. They are simultaneously simple and rich
in their structure, as we shall see. The goal of this section is to introduce the notion of Teichmüller representatives,
which are of fundamental importance in arithmetic geometry.
As before, unless otherwise specified, k will always denote a field and v a discrete valuation on k.
Definition 5.1. The pair (k, v) is called a local field if v is complete and the residue field F v = O v /mv is a finite field.
Thus, F v = Fq where Fq is a field of prime characteristic p (so that q is a p th power). In this case, we call (k, v) a
p-field.
Corollary 5.1. Let (k, v) be a local p-field with residue field Fq . Let mnv be any non-zero ideal of O v . Then O v /mnv is a
finite ring of size q n .
Proof. By assumption, O v /mv ∼
= Fq , so that [O v : mv ] = q. Since indices of ideals are multiplicative, the result is
proved.
Theorem 5.1. Let (k, v) be a local p-field with residue field F v ∼
= Fq . For any x ∈ O v , the limit
ω(x) := lim x q
n
n→∞
exists and is an element of O v . This defines a map ω : O v → O v , which has the following properties for all x, y ∈ O v :
1. ω(x) − x ∈ mv .
2. ω(x)q = ω(x).
3. ω(x y) = ω(x)ω(y).
Remark 9. A common notation, which we will adopt, is that a ≡ b (mod mv ) if and only if a − b ∈ mv for a, b ∈ O v .
7
³ n ´∞
Proof. First, we will inductively prove a set of congruences. This will prove that the sequence x q
n=1
in O v is
Cauchy. Since k is complete (being a local field) and O v is closed in k, this will prove that the above sequence has a
limit in O v .
To this end, we prove that for all n ∈ N,
n
xq ≡ xq
n−1
(mod mnv ).
The base case follows by the "freshman’s dream", since O v /mv is a finite field of size q and q is a p th power. Now
m
q
suppose the congruence holds for some m ∈ N, so that there is some y ∈ mm
= xq
v such that x
m−1
+ y. The Binomial
Theorem gives
x
q m+1
³
= x
q m−1
+y
´q
à !
à !
q
q−1
X
X q i q m−1 q−i
q i q m−1 q−i
qm
=
x
y
=x +
x
y
.
i =0 i
i =0 i
We claim that in the sum in the last equality, each binomial coefficient is divisible by p. Indeed, the i = 0 case is
clear since q is a p th power. Now for 0 < i ≤ q − 1,
!
à !
Ã
q
q q −1
=
i
i i −1
is divisible by p for the same reason. This proves that
x
q m+1
¡q ¢
i
y q−i is contained in mm+1
, so the above sum expresses
v
in the desired form.
n
Now for n ∈ N, observe that if we write x q − x q
n−1
³ n
´
n−1
= y for some y ∈ mnv , then n = v x q − x q
by an earlier
remark
³ n ´∞ on the alternative view of valuations. In particular, the valuations of differences in the terms of the sequence
is unbounded, i.e. this sequence is Cauchy. As noted above, this implies that it converges to some element
xq
n=1
ω(x) of O v .
n
Continually reducing the above congruence to lower powers of mv , we see that x q = x + y for some y ∈ mv , so
that
n
lim x q = lim x + lim y
n→∞
n→∞
n→∞
ω(x) = x + y
ω(x) ≡ x (mod mv ).
Continuity of the ring operations in O v gives
ω(x)q = lim x q
n+1
n→∞
n
= lim x q = ω(x)
n→∞
and
n
n
ω(x y) = lim x q y q = ω(x)ω(y).
n→∞
This completes the proof.
Definition 5.2. The map ω : O v → O v defined in the previous theorem is called the Teichmüller map of v. For any
x ∈ O v , the element ω(x) ∈ O v is called the Teichmüller representative of x.
The Teichmüller map has many important properties and is a fundamental construction which underlies much
of modern algebraic geometry and arithmetic geometry. We scratch the surface of these properties in the following
result.
8
Theorem 5.2. Let (k, v) be a local p-field with residue field F v ∼
= Fq , and define the two sets
V = {x ∈ k | x q−1 = 1},
A = V ∪ {0} = {x ∈ k | x q = x}.
Then A is a complete set of representatives of the quotient F v = O v /mv which contains 0; V is the set of all (q − 1)st
roots of unity in k; and the natural ring homomorphism O v → O v /mv induces an isomorphism of multiplicative
groups,
∗
V∼
= Fv .
In particular, V is a cyclic group of order q − 1.
Proof. Let A 0 = ω(O v ) = {ω(x) | x ∈ O v }. Then we claim that A 0 = A. Indeed, the previous theorem showed that
ω(x)q = ω(x) for all x ∈ O v , so A 0 ⊆ A. Further, we know that ω(x) ≡ x (mod mv ), so each residue class of O v /mv
contains at least one element of A 0 . But as a polynomial in k[x], there are at most q roots of x q − x, and there are q
distinct residue classes in O v /mv ∼
= Fq . Since A 0 ⊆ A, this forces A = A 0 , and also shows that A has q elements. Clearly
0 = ω(0) ∈ A, so the assertion about A is proved. The assertions about V follow from the fact that ω is multiplicative,
completing the proof.
Example 5.1. As mentioned in Example 3.1, the p-adic integers Zp can be viewed as formal power series of the form
∞
X
a i p i = a 0 + a 1 p + a 2 p 2 + ...
i =0
In fact, these power series are more than formal, as they can be shown to converge with the p-adic valuation. But the
question remains: where should the coefficients a i be taken from? The natural choice is from the set of least residue
classes mod p, i.e. a i ∈ {0, 1, ..., p − 1}. However, this leads to trouble when trying to describe the ring operations in Zp
explicitly.
P
P
If a = a i p i and b = b i p i are two p-adic integers with a i , b i chosen as above, how can we describe the sum
c := a + b in terms of its coefficients c i ? A technically correct description, obtained by truncating at each power of p, is
c0 ≡ a0 + b0
(mod p)
c 0 + c 1 p ≡ (a 0 + b 0 ) + (a 1 + b 1 )p
(mod p 2 )
c 0 + c 1 p + c 2 p 2 ≡ (a 0 + b 0 ) + (a 1 + b 1 )p + (a 2 + b 2 )p 2
(mod p 3 )
and so on. A similar description of multiplication in terms of Cauchy products can be written. But notice that, for
example, c 0 may not be in the representative set {0, 1, ..., p − 1}, even if a 0 and b 0 are. The same issue arises for the other
coefficients. In summary, we do not obtain a closed-form description of the ring operations on Zp with these choices of
coefficients.
The better choice of representatives is the set of Teichmüller representatives of the a i and b i . That is, we replace a i in
P
the power series a = a i p i with ω(a i ), which can be viewed as the unique (p − 1)st root of unity which is congruent to
a i (mod p), and similarly for b i . This can be shown to produce the implicit relations
c0 ≡ a0 + b0
p
p
(mod p)
p
c0 + c1 p ≡ a0 + a1 p + b0 + b1 p
(mod p 2 )
9
p2
p
p2
p
p2
p
c0 + c1 p + c2 p 2 ≡ a0 + a1 p + a2 p 2 + b0 + a1 p + b2 p 2
(mod p 3 )
etc. Unlike before, this implicit definition of addition in Zp is closed-form in the sense that each coefficient c i of the
sum a + b is given in terms of Teichmüller representatives. This leads to the very general construction of so-called Witt
vectors, an exposition on which can be found in [5].
10
6 F URTHER W ORK AND C ONCLUSION
There are several important milestones beyond the Teichmüller map that can be reached in class field theory. One
can generalize the notion of ramification index and inertial degree to extensions of valuation fields. There are
important results analogous to our work in this course, and the application of these methods to infinite extensions
defines class field theory to a large extent. Even more generally, one can study profinite groups - a type of topological
group which encompasses all Galois groups.
As mentioned in the introduction, the first important theorem in class field theory is the Kronecker-Weber
Theorem, which classifies Abelian extensions of the rational numbers. From here one can prove the existence of the
so-called Hilbert class field, which was alluded to during our course. This is an extension of a number field with
several marvellous properties that captures information about all unramified extensions.
In summary, class field theory is an exceedingly deep and rich theory with broad applications to the rest of number
theory and arithmetic geometry.
11
R EFERENCES
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Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.), 116: 601–655. JFM 38.0501.01
[3] Iwasawa, K. (1982). Local class field theory. Oxford, UK: Oxford University Press.
[4] Krull, W. (1939). Beiträge zur Arithmetik kommutativer Integritätsbereiche. VI. Der allgemeine Diskriminantensatz. Unverzweigte Ringerweiterungen. Mathematische Zeitschrift, 45 (1): 1–19. doi:10.1007/BF01580269
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https://arxiv.org/abs/1409.7445
[6] Roe, D. (2013). Introduction to arithmetic geometry. Personal Collection of D. Roe, MIT, Cambridge, MA. [PDF].
Retrieved from https://math.mit.edu/classes/18.782/LectureNotes4.pdf
[7] Rudin, W. (1953). Principles of mathematical analysis (pp. 11-12). New York, NY: McGraw-Hill.
[8] Weissman, M.H. Lectures on algebraic number theory. Personal Collection of M. H. Weissman, UC Santa Cruz,
Santa Cruz, CA. [PDF]. Retrieved from https://people.ucsc.edu/ weissman/Math222A/MHWNotesW3.pdf
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