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THE P-ADIC NUMBERS AND BASIC THEORY OF
VALUATIONS
ANDREW DING
Abstract. In this paper, we aim to study valuations on finite extensions of
Q. These extensions fall under a special type of field called a global field. We
shall also cover the topics of the Approximation Theorems and the ring of
adeles, or valuation vectors.
Contents
1. Valuations
2. Types of Valuations
3. Ostrowski’s Theorem
4. Weak Approximation Theorem
5. Finite Residue Field
6. Haar Measure
7. Normed Spaces
8. Extensions of (Normalized) Valuations
9. Global Fields
10. Restricted Topological Product
11. Ring of Adeles
12. Strong Approximation Theorem
13. Conclusion
Acknowledgments
References
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The paper will assume knowledge over several topics. In particular, we will assume basic familiarity with point-set topology, algebraic number theory, topological
groups, and the notion of Haar Measure.
Most of the material comes from “Algebraic Number Theory” by Cassels and
Fröhlich [1].
1. Valuations
Definition 1.1. A valuation on a field k is a function | · | : k → R≥0 satisfying
these axioms:
(1) |α| = 0 ⇐⇒ α = 0
(2) |αβ| = |α||β|
(3) There is a constant C ≥ 1 such that |1 + α| < C whenever |α| ≤ 1.
1
2
ANDREW DING
Definition 1.2. The trivial valuation on k is the following:
(
0 if α = 0,
|α| =
1 if otherwise.
Every field k has the trivial valuation. If k has a nontrivial valuation, we may say
k is a valued field.
We will usually not consider this valuation in future theorems.
Using property (2), we have
|1| = |1| · |1| =⇒ |1| = 1.
Further, given ω ∈ k, if there exists n ∈ N such that ω n = 1, then |ω| = 1 for
similar reasons. We have proved the following.
Corollary 1.3. For finite fields, the only valuation is the trivial one.
Definition 1.4. Two absolute values | · |, | · |0 on Q are equivalent if there exists
some constant c > 0 such that
|x|0 = |x|c .
Note that if |x| is a valuation, then |x|c is as well. Also, this is an equivalence
relation.
Remark 1.5. Every valuation is equivalent to a valuation with C = 2. For these
valuations, one can prove the triangle inequality: for all α, β ∈ k
|α + β| ≤ |α| + |β|
Conversely, given a function satisfying (1), (2), and the triangle inequality, it is
trivial that (3) is satisfied with C = 2.
Remark 1.6. One may wonder from the above argument, why (3) is used instead
of the triangle inequality. We do this because, in the future, we want to consider
the square of the complex absolute value as a valuation.
2. Types of Valuations
We will cover two properties of a valuation: discreteness and the archimedean
property.
Definition 2.1. A valuation | · | on k is called discrete if there exists a δ > 0 such
that if α ∈ k satisfies
1 − δ < |α| < 1 + δ,
then |α| = 1.
One equivalent condition is that the group consisting of ln α is a discrete (additive) subgroup of R. Such a group is necessarily isomorphic to Z.
Definition 2.2. Given a discrete valuation | · | on k, the value group is the
(multiplicative) group
S := {|α| ∈ R≥0 : α ∈ k} ⊆ R≥0 .
From the above, it is clear that S = {cm : m ∈ Z} for some 0 < c < 1. If |α| = cm ,
then we call m the order of α.
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS
3
Definition 2.3. A valuation | · | on k is said to be nonarchimedean if we can
take C = 1 in the definition of a valuation, that is, for all α, β ∈ k,
|α + β| ≤ max{|α|, |β|}.
Definition 2.4. A class of equivalent valuations on k is called a place or prime.
We will call the nonarchimedean primes finite, and the archimedean primes infinite.
We make the following remark for future use.
Remark 2.5. If |α| < |β|, then |α + β| = |β|. This is because
|α| = |(α + β) − β| ≤ max{|α + β|, |β|}.
Definition 2.6. Suppose | · | is a nonarchimedean valuation on k. The set
o := {α ∈ k : |α| ≤ 1}
is a ring, called the valuation ring. It has a unique maximal ideal
p := {α ∈ k : |α| < 1}.
We have two statements we can say about valuations.
Lemma 2.7. A nonarchimedean valuation | · | is discrete if and only if p is a
principal ideal.
Lemma 2.8. A necessary and sufficient condition for a valuation | · | on k to be
nonarchimedean is that |n| ≤ 1, where n is an element of the subring generated by
1 in k (that is, the intersection of all subrings of k containing 1).
Proof. Suppose | · | is an nonarchimedean valuation. For any n, we have
|n| = |1 + · · · + 1| ≤ |1| = 1.
Conversely, suppose |n| ≤ 1 for all n in the subring generated by 1. Assume WLOG
that |x| ≥ |y|. Using the binomial theorem,
n X
n
|x|n−k |y|k ≤ (n + 1)N |x|n .
|x + y|n =
k
k=0
Therefore,
|x + y| ≤ ((n + 1)N )1/n |x| = ((n + 1)N )1/n max{|x|, |y|}.
By letting n → ∞, we obtain the desired inequality.
Corollary 2.9. If char k = p (p nonzero), then all valuations on k are nonarchimedean.
Proof. The ring generated by 1 in k is a finite field. Every element α satisfies
|α| = 1 (Corollary 1.3), so we can apply the above lemma.
We end this subsection with an important example. Let K be an algebraic
number field with ring of integers o. As we have seen in Definition 2.6, a nonarchimedean valuation on K determines a prime ideal of its ring of integers R. In
fact, if two valuations are equivalent, then they produce the same prime ideal, so
in fact, nonarchimedean places determine prime ideals.
Less obviously, a prime ideal p ⊆ o determines a valuation. By a theorem in
algebraic number theory, o is a Dedekind Domain. Therefore, the localization of o
4
ANDREW DING
at p is a Discrete Valuation Ring, which gives a valuation on the field of fractions
of R, that is, K.
3. Ostrowski’s Theorem
We shall state a necessary theorem before starting this section, usually attributed
by Ostrowski.
Theorem 3.1. Any nontrivial absolute value on Q is equivalent to either the Euclidean absolute value or a p-adic absolute value.
Proof. Cassels and Fröhlich, p.46-47 (Chapter 2, Section 3).
Now, let k0 be a field, and k0 (t) be a transcendental extension. If p = p(t) ∈ k0 [t]
is an irreducible polynomial, then we can define a valuation
(p(t))a u(t) = c−a ,
(3.2)
v(t) p
where c < 1 is fixed, u(t), v(t) ∈ k0 [t] and p(t) - u(t) and p(t) - v(t).
In addition, there is the non-archimedean valuation | · |∞ given by
u(t) deg v−deg u
(3.3)
.
v(t) = c
∞
This gives rise to an analogy between Q and k0 (t), which is not perfect, since the
absolute value “at infinity” for Q is archimedean, but not so for k0 (t). However,
we do have the following results, which are proved similarly as above.
Lemma 3.4. The only nontrivial valuations on k0 (t) that are trivial on k0 are
given by (3.2) and (3.3) above.
Corollary 3.5. Let F be a finite field. The only nontrivial valuations on F (t) are
given by (3.2) and (3.3) above.
Proof. Recall that a finite field only has the trivial valuation (Corollary 1.3).
If k is a valued field, we can give it a topology with basis the open balls
Sd (α) = {β ∈ k : |β − α| < d},
where α ∈ k and d > 0. It turns out that equivalent valuations give the same
topology. In addition, it is true that k is a topological field under this topology;
that is, the sum, product, and inversion operations are all continuous.
Lemma 3.6. Suppose two valuations | · |, | · |0 on the same field k induce the same
topology. Then they are equivalent.
Proof. Let α ∈ k. We have |α| < 1 ⇐⇒ limn→+∞ αn = 0. It follows that
|α| < 1 ⇐⇒ |α|0 < 1. Taking inverses, we have |α| > 1 ⇐⇒ |α|0 > 1. By
elimination, we can conclude that |α| = 1 ⇐⇒ |α|0 = 1.
Let β, γ ∈ k be nonzero, and let m, n ∈ Z. By letting α = β m γ n , we have
m ln |β| + n ln |γ| >, <, or = 0 ⇐⇒ m ln |β|0 + n ln |γ|0 >, <, or = 0,
respectively. It follows that
ln |β|
ln |γ|
=
ln |β|0
ln |γ|0
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS
5
4. Weak Approximation Theorem
This result shows that inequivalent valuations are, in some sense, unrelated to
one another. There will be a Strong Approximation Theorem, which is why the
following is a lemma.
Lemma 4.1. Let | · |1 , . . . , | · |n be inequivalent nontrivial valuations over a field k.
Then given any αj ∈ k(1 ≤ j ≤ n), and > 0, there exists a ξ ∈ k that satisfies
|αi − ξ|i < simultaneously.
The Weak Approximation Theorem is related to the Chinese Remainder Theorem. If we let k = Q, and have n p-adic valuations | · |pi , for any n-tuple of
rational numbers (a1 , . . . , an ), we can find a x ∈ Q such that |x − ai |pi < , that is,
x − ai ≡ 0 mod pki for arbitrary k ∈ Z.
Proof. By Proposition 3.6, there exists an a ∈ k such that |a|1 < 1 and |a|n ≥ 1.
Similarly, there exists a b ∈ k such that |b|n < 1 and |b|1 ≥ 1. Letting c = b/a, we
have |c|1 > 1 and |c|n < 1.
Now, we prove by induction on n that there exists z ∈ k such that
|z|1 > 1 and |z|j ≤ 1 for j = 2, . . . , n.
Above, we have done this for n = 2, so assume there exists a z ∈ k such that
|z|1 > 1 and |z|j ≤ 1 for j = 2, . . . , n − 1.
Suppose |z|n ≤ 1. Suppose c ∈ k satisfies |c|1 > 1 and |c|n < 1. For j = 2, . . . , n −
1, |c|j might be large, but there exists a sufficiently large m such that z m c satisfies
|z m c|j < 1 for j = 2, . . . , n. Therefore, set z 0 = z m c. Now suppose |z|n > 1.
zm
We claim that the sequence defined by am = 1+z
m converges to 1 with respect to
| · |1 and | · |n , but to 0 with respect to | · |j for j = 2, . . . , n − 1. This is because
−m m = 1 = z
≤ |z|−m → 0 for j = 1, n
1 − z
j
m
m
−m
1+z j
1+z j
1+z
j
m z
m
1 + z m ≤ |z|j → 0 for j = 2, . . . , n − 1
j
Therefore, we can set z 0 = am c by the same reasoning as the |z|n ≤ 1 case. Induction complete.
Now we have a z such that |z|1 > 1 and |z|j < 1 for j = 2, . . . n. For the same
zm
reason as the paragraph above, bm = 1+z
m converges to 1 with respect to | · |1 and
to 0 with respect to | · |j for j 6= 1. By symmetry, for i = 1, . . . , n, we can construct
zi that is “close to 1” with respect to | · |i and “close to 0” with respect to | · |j
where j 6= i. Set
ξ = α1 z1 + · · · + αn zn .
The element ξ will satisfy |αi − ξ|i < for all i simultaneously.
5. Finite Residue Field
Let k be a field with nonarchimedean valuation | · |. We recall
o := {α ∈ k : |α| ≤ 1}
6
ANDREW DING
is an integral domain. In addition, given ∈ k, we have || = 1 if and only if ∈ o× .
Finally,
p := {α ∈ k : |α| < 1}
is obviously an ideal. Since it contains all non-unit elements, it is a maximal ideal,
so that o/p is a field. We are interested in the case where the residue field o/p is
finite.
If we suppose further that | · | is discrete, then we can use the theory of discrete
valuation rings. We have that p is a principal ideal (π) for some π, and that every
element α ∈ o can be expressed as α = π k for some k ∈ N≥0 , where is a unit.
Definition 5.1. Let α = π k ∈ o. The order of α is defined to be v.
This is well-defined since if p = (π 0 ), then π/π 0 and π 0 /π are both units, in
particular, if α = π k , then α = 0 (π 0 )k for some unit 0 .
Now consider the completion of k with respect to | · |, and call it k̄. Define ō, p̄
with respect to k̄. We have ō/p̄ ∼
= o/p and the following lemma:
Lemma 5.2. Suppose k is complete with respect to a discrete, nonarchimedean
valuation | · | and suppose the residue field o/p is finite. Then every element α ∈ o
has a unique representation
∞
X
α=
aj π j ,
j=0
where each aj ∈ S, where S is a set of representatives for o/p.
To be formal, the infinite series above is the limit of the sequence of partial sums.
Proof. Since p is a subgroup of o, there is a unique element a0 ∈ S such that α−a0 ∈
p. Now α = a0 + πα1 , where α1 = π −1 (α − a0 ) ∈ o. Now we can define a1 in the
same way, and repeat for a2 , a3 , etc.
Theorem 5.3. Suppose k is complete with respect to a discrete, nonarchimedean
valuation | · | and suppose the residue field o/p is finite. Then o is compact with
respect to the metric topology induced by | · |.
Proof. Proof by contradiction. Let Oλ (λ ∈ Λ) be a collection of open sets. Consider
the finite collection of open sets {ai + πo}, where we take all ai ∈ S. Since o is not
compact, one of the above sets is not covered by finitely many Oλ ; call it b0 + πo.
Proceeding inductively, considering {b0 + ai π 2 o}, there exists a b1 such that b0 +
b1 π + π 2 o is not finitely covered, and so on.
Define α = b0 + b1 π + · · · . By definition of cover, there exists a λ such that α ∈
Oλ . Since the cosets α+π n o are open balls that form a neighborhood basis for α, we
have α+π n o ⊆ Oλ for some n, which is a contradiction to b0 +· · ·+bn−1 π n−1 +π n o =
α + π n not having a finite cover.
Corollary 5.4. k is locally compact.
It is worth mentioning that the converse is true, i.e., if k is a field that is locally
compact with respect to the nonarchimedean valuation | · |, then k is complete, the
residue field is finite, and the valuation is discrete.
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS
7
6. Haar Measure
Now is the appropriate time to talk about Haar measure. We will not go into
much detail, but it is a very useful tool. In fact, we will use the properties of Haar
measure to prove the Product Formula in a later subsection.
Definition 6.1. A Haar measure µ on a locally compact topological group is a
translation-invariant measure such that every open set can be covered by open sets
with finite measure.
Given a locally compact topological group, measure theory tells us that there
exists a Haar measure, and it is unique up to a scaling factor.
Remark 6.2. It follows by definition that every compact set has finite measure.
Conversely, a translation-invariant measure such that every compact set has finite
measure implies the condition in Definition 6.1 using local compactness.
Let k be a field with nonarchimedean valuation | · |, with o/p finite (and S a set
of representatives, π a generator for p). Let us determine the Haar measure on k + .
Suppose that
µ(α + π v o) = µv ,
for some constant µv . We have
µ(α + π v o) =
[
(α + π v aj + π v+1 o),
1≤j≤|S|
where aj ∈ S are representatives of o/p, and |S| is the cardinality of S. Therefore,
µ(α + π v ) = µv = |S|µv+1 .
We will normalize the measure as follows. We require that
µ(o) = 1.
This implies that µv = |S|−v . This motivates the notion of a normalize d valuation.
Definition 6.3. Let k be a field with nonarchimedean valuation | · |, and |o/p| =
P < ∞. We say that | · | is normalized if
|π| = P −1 ,
where p = (π).
Theorem 6.4. Suppose that k is complete with respect to the normalized valuation
| · |. Then
µ(α + βo) = |β|,
where µ is the Haar measure on k + normalized by µ(o) = 1.
The above theorem will motivate a similar notion of normalization for nonarchimedean valuations.
8
ANDREW DING
7. Normed Spaces
Definition 7.1. Let k be a valued field with respect to | · |, and V be a vector
space over k. A function k · k : V → R is called a norm if for all a, b ∈ V, α ∈ k,
(1) kak > 0 if a 6= 0
(2) ka + bk ≤ kak + kbk
(3) kαak = |α|kak.
If these properties hold, then V is called a normed space.
Definition 7.2. Two norms k · k, k · k0 on V are equivalent if there exist constants
c1 , c2 such that for all a ∈ V ,
kak ≤ c1 kak0 and kak0 ≤ c2 kak.
This is an equivalence relation on the set of norms.
Lemma 7.3. Suppose k is complete with respect to |·|, and V is a finite-dimensional
vector space over k. Then any two norms on V are equivalent.
Proof. Let a1 , . . . , an be a basis for V . We define the norm k · k0 as follows:
n
X
ξj aj = max |ξj |.
j=1
Our aim to show that any norm k · k is equivalent to k · k0 . We have
X
X
X
X
n
n
X
n
n
n
kaj k.
ξ
a
ka
k
=
|ξ
|ka
k
≤
(max
|ξ
|)
ξ
a
≤
j j
j
j
j
j
j j
j=1
j=1
j=1
j=1
j=1
0
Pn
Therefore, set c1 = j=1 kaj k.
For c2 , we will use induction on n. The n = 1 case is obvious; pick {1} as the
basis, and for any norm k · k,
kak = |a|k1k = |a|,
so there is only one norm, determined by the valuation on k.
For the inductive step, wePuse proof by contradiction. Thus assume that for
n
every > 0, there exists b = j=1 ξj aj such that
kbk ≤ max |aj |.
Observe that such a must be nonzero. Therefore, by symmetry, assume that
max |ζj | = |ζn | and, by homogeneity and division, assume that ζn = 1.
For each m = 1, 2, · · · , set = 1/m, and use the above assumption to obtain
Pn−1
bm = j=1 ξj,m aj + an such that
max |ζj |
1
=
→0
(m → ∞).
m
m
Therefore, it is a convergent sequence with respect to k · k, in particular, it is a
Cauchy sequence:
X
n
kbl − bm k = (ζj,l − ζj,m )aj (l, m → ∞).
→0
j=1
kbm k ≤
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS
9
Now we use the induction hypothesis. Suppose all norms on the n − 1-dimensional
vector space spanned by a1 , . . . , an−1 are equivalent to k · k0 . Then for each j,
|ζj,l − ζj,m | → 0
(l, m → ∞).
By completeness of k, there exists a ζj∗ such that
|ζj,m − ζj∗ | → 0
(m → ∞).
So we have
n
n
n
X
X j,m
X ∗
|ξj∗ − ξjj,m |kaj k → 0
(m → ∞),
ξj aj + an ξj aj + an ≤ +
j=1
j=1
j=1
Pn
which contradicts positive-definiteness (as we said earlier, j=1 ξj∗ aj +an 6= 0). 8. Extensions of (Normalized) Valuations
The theorems in this section will not be proven. “Algebraic Number Theory”
by Cassels and Fröhlich [1] (Chapter 2) is recommended, but they cite “Theory of
Algebraic Numbers” by E. Artin on occasion.
Throughout this subsection, suppose L/K is a finite field extension, and k · k
and | · | are valuations on L, K respectively.
Definition 8.1. We say that k · k extends | · | if kak = |a| for all a ∈ K.
One should note that every valuation of L is the extension of some (possibly
trivial) valuation on K. To see this, restrict k · k to K.
We begin with the simpler situation: when k is complete.
Theorem 8.2. Suppose K is a field that is complete with respect to | · |, and L is
a finite extension of degree N = [L : K]. Then there is precisely one extension of
k · k to K, namely
kak = |NormL/K (a)|1/N .
We will not prove this here, but uniqueness requires Lemma 7.
If we do not assume K is complete with respect to | · |, the situation becomes
more complicated.
Theorem 8.3. Let L be a separable extension of K of finite degree N = [L : K].
Then there are at most N extensions of a valuation | · | on K to L, call them
k · kj for 1 ≤ j ≤ J. Let Kv be the completion of K with respect to | · |, and for
each j, let Lj be the completion of L with respect to k · kj . Then
M
Kv ⊗K L ∼
Lj .
=
1≤j≤J
Now let us focus on normalized valuations. Suppose k is a valued field. We have
three cases to consider:
(1) | · | is nonarchimedean with finite residue field.
(2) The completion of k with respect to | · | is R.
(3) The completion of k with respect to | · | is C.
10
ANDREW DING
We have already covered the first case (Definition 6.3). In the second case, | · | is
normalized if it is the ordinary absolute value, and in the third case, |·| is normalized
if it is the square of the complex absolute value.
The reasoning behind this is so, for every a ∈ k, and measurable set E, we have
that µ(aE) = |a|µ(E) for any Haar measure µ, and the normalized valuation | · |
(cf. Theorem 6.4).
Again, we have two results, and the simpler case requires k to be complete.
Lemma 8.4. Suppose K is complete with respect to a normalized valuation | · | and
let L be a finite extension of K of degree N = [L : K]. Then for any a ∈ L,
kak = |NormL/K (a)|,
where k · k is a normalized extension, equivalent to the unique extension of | · | to L
(see Theorem 8.2).
If we do not assume K is complete, then we have the following:
Theorem 8.5. Suppose K has normalized valuation | · |, and let L be a finite
extension of K. Then for any a ∈ L,
Y
kakj = |NormL/K (a)|,
1≤j≤J
where the k · kj are the normalized valuations equivalent to the finitely many extensions of | · | to L (see Theorem 8.3).
9. Global Fields
Definition 9.1. A global field is either a finite extension of Q or a finite, separable
extension of F (t), where F is a finite field.
This paper focuses on the algebraic number case, that is, finite extensions of Q.
The following lemma will be needed for the definition of a principal adele in the
next section.
Lemma 9.2. Let k be a global field, and α ∈ k be nonzero. Then there are only
finitely many inequivalent valuations | · | of k such that
|α| > 1.
Proof. We have shown this for Q and F (t) with Ostrowski’s Theorem (Theorem
3.1).
Suppose k is a finite extension of Q, so α satisifies some polynomial
αn + a1 αn−1 + · · · + an = 0,
for some n and coefficients aj ∈ Q. If | · | is a nonarchimedean valuation, then
|α|n = | − a1 αn−1 − · · · − an |
≤ max{1, |α|n−1 } max{|a1 |, . . . , |an−1 |}
We know that either |α| ≤ 1 or |α|n−1 > 1. In the second case, we can divide the
above equation by |α|n−1 to obtain |α| ≤ max{|a1 |, . . . , |an−1 |}. Combining the
two:
|α| ≤ max{1, |a1 |, . . . , |an−1 |}.
Every valuation on k must be an extension of some valuation on Q (via restriction).
Consider the RHS. There are only finitely many valuations on Q that would make
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS
11
the RHS greater than 1, and for each of those, there are only finitely many extensions (Theorem 8.3) to valuations on k. In addition to the fact that there are only
finitely many archimedean valuations, we conclude that, in total, there are only
finitely many valuations on k such that |a| > 1.
10. Restricted Topological Product
The following will be necessary for the next subsection.
Definition 10.1. Let Ωλ (λ ∈ Λ) be a family of topological spaces and for almost
all (i.e. all but finitely many), λ, let Θλ ⊆ Ωλ be open sets.
Let Ω be the space whose points are sets α = {αλ }λ∈Λ such that αλ ∈ Ωλ for
every λ and αλ ∈ Θλ for almost all λ.
We give Ω a topology by defining a basis of open sets, each of the form
Y
Γλ ,
where Γλ ⊆ Ωλ are open for all λ, and Γλ = Θλ for almost all λ.
Lemma 10.2. Let S be a finite subset of Λ. Let ΩS be the set of α such that αλ ∈
Θλ (λ ∈
/ S), that is,
Y
Y
ΩS =
ΩS
ΘS
λ∈S
λ∈S
/
Then ΩS is an open set, and the topology induced as a subspace of Ω is the same
as the product topology.
We remark that the restricted topological product relies on the Θλ as a whole,
but not on the individual level.
Remark 10.3. Let Θλ ⊆ Ωλ be open sets, defined for almost all λ, and suppose
Θλ = Θ0λ for almost all λ. Then the restricted product of the Ωλ with respect to
Θλ is canonically isomorphic to that with respect to Θ0λ .
Lemma 10.4. Suppose that Ωλ are locally compact and Θλ are compact. Then Ω
is locally compact.
Proof. Let S ⊆ Λ be finite. As in Lemma 10.2, each ΩS is a product of an arbitrary
number of compact sets and finitely many locally compact sets, hence is locally
compact. In addition, each ΩS is open, in particular, it has nonempty interior, and
Ω = ∪S XS . Hence, every point x has a compact neighborhood, so Ω is locally
compact.
We will also want the following notion of measure on the restricted topological
product. This is by no means a rigorous definition, but rather its most basic
property.
Definition 10.5. For all λ ∈ Λ, assume that measures µλ exist on Ωλ with
µλ (Θλ ) = 1 whenever Θλ are defined. We define the product measure µ on
X as follows. The basis of measurable sets is
Y
Mλ ,
λ∈Λ
12
ANDREW DING
where Mλ ⊆ Ωλ each have finite measure (with respect to µλ ), and Mλ = Θλ for
almost all λ. We define
!
Y
Y
µ
Mλ =
Mλ .
λ∈Λ
λ∈Λ
By definition of “almost all”, we are taking an infinite product, with only finitely
many nonidentity elements, so the product is well-defined.
11. Ring of Adeles
Let k be a global field. For each valuation | · |v on k, let kv be the completion of
k with respect to | · |v . If | · |v is nonarchimedean, then let ov be its valuation ring.
Definition 11.1. The ring of adeles (or ring of valuation vectors), Ak is the
topological ring whose underlying space is the restricted topological product of kv
(for all valuations v), with respect to ov (for nonarchimedean valuations v). We
define addition and multiplication componentwise:
xv + yv = (x + y)v and xv yv = (xy)v .
One can easily check that addition and multiplication are well-defined. It is also
true that addition and multiplication are continuous with respect to the restricted
product, so it is indeed a topological ring. Lemma 10.4 implies Ak is locally compact, as kv are locally compact (Corollary 5.4), and ov are compact (Theorem 5.3).
The only archimedean completions are either R or C, both of which are locally
compact.
Consider the homomorphism k ,→ Ak , sending α to the adele whose components
are each α. This is well-defined by Lemma 9.2, and is continuous (consider the
preimage of basis elements). The map is injective since k ,→ kv is injective.
Definition 11.2. The image of the above map k ,→ Ak is called the ring of
principal adeles.
We may implicitly identify k with the principal adeles when we write k ⊆ Ak .
We will also need the following result, which is proven in Cassels and Fröhlich
p.64 (Chapter 2, Section 14).
Lemma 11.3. Let A+
k denote the topological group obtained from the additive structure on Ak , and suppose K is a finite separable extension of k. Then
+
+
A+
K = Ak ⊕ · · · ⊕ Ak
In this group, the additive group K
· · · ⊕ k+ .
+
⊆
([K : k] summands)
A+
K
is mapped isomorphically onto k + ⊕
Theorem 11.4. The global field k is discrete in Ak , and the quotient Ak /k is
compact in the quotient topology.
Proof. The above lemma, substituting k for K and Q (or F (t)) for k shows that we
only need to prove it for Q+ . Using properties of topological groups, it is enough
to find an open set U containing 0 but no other element of Q+ . Let
U = {{xv }v ∈ A+
Q : |x∞ |∞ < 1 and |xp |p ≤ 1 for all p,
where | · |p , | · |∞ are the normalized p-adic and Euclidean absolute values on Q. If
α ∈ Q ∩ U , then the second condition implies α ∈ Z, and then the first condition
implies α = 0.
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS
13
+
+
To show A+
Q /Q is compact, define the compact set W ⊆ AQ to be
W = {{xv }v ∈ A+
Q : |x∞ |∞ ≤ 1/2 and |xp |p ≤ 1} for all p,
We want to show that every adele y = {yv }v can be expressed in the form
y = a + x, where a ∈ Q, x ∈ W,
+
which is sufficient, since W would map surjectively onto A+
Q /Q . Now fix an adele
+
y = {yv }v ∈ AQ . Using Weak Approximation Theorem (Lemma 4.1), we can define
zp
rp = np with zp ∈ Z, np ∈ N such that |yp − rp |p ≤ 1.
p
Since y is an adele, we can also impose
rp = 0 for almost all p.
P
∈ Q is well-defined, and since | · |p is nonarchimedean,


X 

|yp − r|p ≤ max |yp − rp |p , −
rq ≤ max{1, 1} = 1.

q6=p 
Therefore, r :=
p rp
p
We can take a s ∈ Z such that
|y∞ − r − s|∞ ≤ 1/2.
(Note that |yp − r − s|p ≤ 1 since |s|p ≤ 1). Now we have y = a + x, where we set
+
+
a := r + s ∈ Q, and x = y − a ∈ W . Therefore, the quotient map A+
Q → AQ /Q
+
+
induces a surjective, continuous map W → AQ /Q . Since W is compact, we can
+
conclude that A+
Q /Q is compact.
This is an interesting result. This suggests that k sits as a lattice inside Ak .
We can compare this to Z sitting inside R, and the fact that R/Z ∼
= S 1 , which
is compact. In fact, the lattice idea is used in the proof in a lemma of the next
subsection.
Corollary 11.5. There is a subset W of Ak defined by inequalities of the form
|xv | ≤ δv , where δv = 1 for almost all v, such that every y ∈ Ak can be expressed as
y = a + x, a ∈ k, x ∈ W.
In particular, Ak = k + W .
Proof. We have done it for Q in the above proof. By Lemma 11.3, for a general
+
field k, the W in the proof determines a subset of Ak ∼
= A+
Q ⊕ · · · ⊕ AQ that has all
the desired properties.
Corollary 11.6. There is a quotient measure on Ak /k, that is induced by the Haar
measure on Ak . Ak /k has finite measure.
Proof. Let U be an open set with finite measure. We can cover Ak with translates
of U , and since Ak /k is compact, the existence of a finite subcover shows Ak /k has
finite measure.
We can now prove the Product Formula in a neat way using the fact that the
Haar measure on the adeles is also the product measure.
14
ANDREW DING
Theorem 11.7. Let k be a global field, and let a ∈ k be nonzero. Let | · |v run
through all the normalized valuations of k. Then |a|v = 1 for almost all v, and
Y
|a|v = 1.
v
Proof. We have seen that, by Theorem 6.4, if xv ∈ Kv , then multiplication by xv
scales the measure by a factor of |xv |v , that is, µ(xv E) = |xv |v µ(E). Therefore,
if
Q
x = {xv }v ∈ Ak , then multiplication by x scales the measure by a factor of |xv |v ,
by the product measure.
However, multiplication by a ∈ k sends k ⊆ Ak to k ⊆ Ak , which givesQa well+
+
defined bijection from A+
to A+
which magnifies the measure by |av |v .
k /k
k /k
Q
+
+
Since (11.6) says the measure on Ak /k is finite, it follows that |av |v = 1.
12. Strong Approximation Theorem
We start with a technical lemma.
Lemma 12.1. There exists a constant C > 0, depending on the global field k, with
the following property: Whenever x = {xv }v ∈ Ak satisfies
Y
|xv |v > C,
v
then there is a nonzero principal adele a ∈ k ⊆ Ak such that for all v,
|a|v ≤ |xv |v .
Proof. Suppose x satisfies the above condition. We note that |x|v = 1 for almost
all v. This is because |x|v ≤ 1 for almost
by definition of adele, and if |x|v < 1
Q all vQ
for infinitely many v, then we have |x|v < pj p1j for an infinite set {pj }, which
goes to 0, a contradiction.
Let c0 be the Haar measure of Ak+ /k + induced from normalized Haar measure
on A+
k , and let c1 be the Haar measure of the set
(
(
)
1
if v is archimedean,
+
2
y = {yv }v ∈ Ak : |yv | ≤
1 if v is nonarchimedean.
We have 0 < c0 < ∞ by compactness (Corollary 11.6), and 0 < c1 < ∞ because
there are only finitely many archimedean valuations. We will show that
c0
C=
c1
gives the desired constant.
Define the set
(
(
)
1
|αv |v if v is archimedean,
+
2
T := t = {tv }v ∈ Ak : |tv | ≤
|αv |v if v is nonarchimedean.
Then T has measure
c1
Y
|αv |v > c1 C = c0 .
v
Note the measure of T is greater than the measure of Ak+ /k + . In the quotient map
Vk+ → Vk+ , there must be a pair of distinct elements
t0 = {t0v } ∈ T and t00 = {t00v } ∈ T such that t0 − t00 ∈ k + .
THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS
15
Then
(
|t0v
−
t00v |v
≤
|t0v | + |t00v | ≤ 2 · 21 |xv |v = |xv |v if v is archimedean,
max{t0v , t00v } ≤ |xv |v if v is nonarchimedean.
Therefore, a := t0 − t00 ∈ k satisfies the requirements of the lemma.
Corollary 12.2. Let v0 be a nontrivial, normalized valuation and suppose we are
given δv > 0 for all v 6= v0 , with δv = 1 for almost all v. Then there is a nonzero
a ∈ k such that
|a|v ≤ δv for v 6= v0
Proof. Choose xv Q
∈ kv with 0 < |x|v ≤ δv , and |xv |v = 1 if δv = 1. We can choose
xv0 ∈ kv0 so that all v |xv |v > C. Then apply the lemma.
We have shown that k is discrete in Ak , but the Strong Approximation Theorem states that if we remove a valuation, something completely different happens.
The Strong Approximation Theorem is a generalization of the Chinese Remainder
Theorem.
Theorem 12.3. Let v0 be any nontrivial, normalized valuation on the global field
k. Let Ak,v0 be the restricted topological product of the kv with respect to the ov ,
except that v runs through all valuations except for v0 . Then k is dense in Ak,v0 .
Proof. K being dense means that given three things: a finite set S of valuations
v 6= v0 , elements xv ∈ kv for all v ∈ S, and an > 0, then there exists an element
b ∈ k such that |b − xv |v < for all v ∈ S and |b|v ≤ 1 for all v ∈
/ S with v 6= v0 .
By Corollary 11.5), there exists a W ⊆ Ak , defined by inequalities of the form
|yv | ≤ δv (where δv = 1 for almost all v such that every z ∈ Ak can be expressed as
z = y + c, y ∈ W, c ∈ k
By Corollary 12.2, there is a nonzero a ∈ k such that
|a|v <
Writing z =
expressed as
1
a
1
for v ∈ S, and |a|v ≤
for v ∈
/ S, v 6= v0 .
δv
δv
· x and multiplying by a, it is shown that every x ∈ Ak can be
x = w + b, w ∈ aW, b ∈ k,
Now, set the components of x to be the given xv if v ∈ S, and 0 elsewhere. It
follows that b = x − w has the required properties.
Now we note that the proof of k discrete in Ak (Theorem 11.4) relies on using
all the valuations v. Recall that in the proof of Theorem 11.4, we found U ⊆ A+
Q
by the elements x such that
|x∞ |∞ < 1 and |x|p ≤ 1,
and the only element of U ∩ Q is 0. However, this relies on the presence of all the
valuations. Intuitively, if we remove a valuation v0 , then we cannot find any open
set V such that V ∩ Q = {0}.
16
ANDREW DING
13. Conclusion
This paper merely scratches the surface of the ring of adeles. There are many
directions one can go from here. First of all, the structure on the ring of adeles in
the algebraic number field case can be used to show Dirichlet’s Unit Theorem and
the finiteness of the ideal class group. Another direction, if the reader is familiar
with harmonic analysis, is Tate’s thesis, where the adeles are further developed.
The reader may wish to research “Global Class Field Theory” for more information
on the subject.
Acknowledgments
I would like to thank Professor May for correcting mistakes and running the
REU for the summer of 2013. I would also like to thank my mentors Chang-Mou
Lim and Daniel Johnstone for explaining several concepts in number theory and
for their guidance on the paper topic. I would also like to thank Daniel Johnstone
again for promptly proofreading several drafts of this paper. All the help is greatly
appreciated.
References
[1] Editors: John William Scott Cassels and Albrecht Fröhlich. “Algebraic Number Theory.”
London Mathematical Society 2010
[2] Author: Jrgen Neukirch, Translator: Norbert Schappacher. “Algebraic Number Theory.”
Springer 1999