The P-Adic Numbers and Basic Theory of Valuations

Home , Discrete valuation ring, Valuation (algebra), Valuation ring

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 1 2. Types of Valuations 2 3. Ostrowski’s Theorem 4 4. Weak Approximation Theorem 5 5. Finite Residue Field 5 6. Haar Measure 7 7. Normed Spaces 8 8. Extensions of (Normalized) Valuations 9 9. Global Fields 10 10. Restricted Topological Product 11 11. Ring of Adeles 12 12. Strong Approximation Theorem 14 13. Conclusion 16 Acknowledgments 16 References 16

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 Fr¨ohlich [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 ωn = 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 + y|n = |x|n−k|y|k ≤ (n + 1)N|x|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 Eu- clidean absolute value or a p-adic absolute value.

Proof. Cassels and Fr¨ohlich, 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

a u(t) −a (3.2) (p(t)) = c , 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) = c . v(t) ∞ 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.

n Proof. Let α ∈ k. We have |α| < 1 ⇐⇒ limn→+∞ α = 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 = βmγn, 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 The- orem. 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, k x − ai ≡ 0 mod pi 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 − m 1, |c|j might be large, but there exists a sufficiently large m such that z c satisfies m 0 m |z c|j < 1 for j = 2, . . . , n. Therefore, set z = z c. Now suppose |z|n > 1. zm We claim that the sequence defined by am = 1+zm 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 z 1 z −m 1 − m = m = −m ≤ |z|j → 0 for j = 1, n 1 + z j 1 + z j 1 + z j

m z m m ≤ |z|j → 0 for j = 2, . . . , n − 1 1 + z j 0 Therefore, we can set z = amc by the same reasoning as the |z|n ≤ 1 case. Induc- tion 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+zm 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 = α1z1 + ··· + αnzn.

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 k element α ∈ o can be expressed as α = π for some k ∈ N≥0, where is a unit. Definition 5.1. Let α = πk ∈ o. The order of α is defined to be v.

This is well-defined since if p = (π0), then π/π0 and π0/π are both units, in particular, if α = πk, then α = 0(π0)k for some unit 0. Now consider the completion of k with respect to | · |, and call it k¯. Define ¯o, p¯ with respect to k¯. We have ¯o/p¯ =∼ 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 j α = ajπ , 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 ∈ −1 p. Now α = a0 + πα1, where α1 = π (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. 2 Proceeding inductively, considering {b0 + aiπ o}, there exists a b1 such that b0 + 2 b1π + π o is not finitely covered, and so on. Define α = b0 + b1π + ··· . By definition of cover, there exists a λ such that α ∈ n Oλ. Since the cosets α+π o are open balls that form a neighborhood basis for α, we n n−1 n have α+π o ⊆ Oλ for some n, which is a contradiction to b0+···+bn−1π +π o = n α + π 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 v µ(α + π o) = µv, for some constant µv. We have

v [ v v+1 µ(α + π o) = (α + π aj + π o), 1≤j≤|S| where aj ∈ S are representatives of o/p, and |S| is the cardinality of S. Therefore,

v µ(α + π ) = µv = |S|µv+1.

We will normalize the measure as follows. We require that

µ(o) = 1.

−v This implies that µv = |S| . 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 , 0 0 kak ≤ c1kak and kak ≤ c2kak. 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 ξjaj = max |ξj|. j=1

Our aim to show that any norm k · k is equivalent to k · k0. We have

n n n n n X X X X X ξjaj ≤ |ξj|kajk ≤ (max |ξj|) kajk = ξjaj kajk.

j=1 j=1 j=1 j=1 j=1 0 Pn Therefore, set c1 = j=1 kajk. 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, we use proof by contradiction. Thus assume that for Pn every > 0, there exists b = j=1 ξjaj 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,maj + an such that max |ζ | 1 kb k ≤ j = → 0 (m → ∞). m m m Therefore, it is a convergent sequence with respect to k · k, in particular, it is a Cauchy sequence:

n X kbl − bmk = (ζj,l − ζj,m)aj → 0 (l, m → ∞).

j=1 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 ∗ j,m ξ aj + an ≤ ξ aj + an + |ξ − ξ |kajk → 0 (m → ∞), j j j j 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 1/N kak = |NormL/K (a)| . 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,

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 n n−1 α + a1α + ··· + an = 0, for some n and coefficients aj ∈ Q. If | · | is a nonarchimedean valuation, then n n−1 |α| = | − a1α − · · · − 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 0 Θλ = Θλ for almost all λ. Then the restricted product of the Ωλ with respect to 0 Θλ is canonically isomorphic to that with respect to Θλ.

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 = ∪SXS. 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 xvyv = (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 Fr¨ohlich p.64 (Chapter 2, Section 14). + Lemma 11.3. Let Ak denote the topological group obtained from the additive structure on Ak, and suppose K is a ﬁnite separable extension of k. Then + + + AK = Ak ⊕ · · · ⊕ Ak ([K : k] summands) + + + In this group, the additive group K ⊆ AK is mapped isomorphically onto k ⊕ · · · ⊕ 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 ∈ : |x∞|∞ < 1 and |xp|p ≤ 1 for all p, AQ 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 +/ + is compact, deﬁne the compact set W ⊆ + to be AQ Q AQ + W = {{xv}v ∈ : |x∞|∞ ≤ 1/2 and |xp|p ≤ 1} for all p, AQ

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 +/ +. Now fix an adele AQ Q + y = {yv}v ∈ . Using Weak Approximation Theorem (Lemma 4.1), we can define AQ zp rp = with zp ∈ Z, np ∈ N such that |yp − rp|p ≤ 1. pnp Since y is an adele, we can also impose

rp = 0 for almost all p. P Therefore, r := p rp ∈ Q is well-deﬁned, and since | · |p is nonarchimedean,  

 X  |yp − r|p ≤ max |yp − rp|p, − rq ≤ max{1, 1} = 1.

 q6=p  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 ∈ , and x = y − a ∈ W . Therefore, the quotient map + → +/ + Q AQ AQ Q induces a surjective, continuous map W → +/ +. Since W is compact, we can AQ Q conclude that +/ + is compact. AQ Q

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 =∼ S1, 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 k = ⊕ · · · ⊕ that has all A AQ AQ 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(xvE) = |xv|vµ(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 gives a well- + + + + Q defined bijection from Ak /k to Ak /k which magnifies the measure by |av|v. + + 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 all v by definition of adele, and if |x|v < 1 Q Q 1 for infinitely many v, then we have |x| < j for an infinite set {p }, which v pj p j goes to 0, a contradiction. + Let c0 be the Haar measure of Ak+ /k induced from normalized Haar measure + on Ak , and let c1 be the Haar measure of the set ( ( 1 ) + 2 if v is archimedean, y = {yv}v ∈ A : |yv| ≤ k 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 c C = 0 c1 gives the desired constant. Deﬁne the set ( ( 1 ) + 2 |αv|v if v is archimedean, T := t = {tv}v ∈ Ak : |tv| ≤ |αv|v if v is nonarchimedean. Then T has measure Y c1 |αv|v > c1C = 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 0 0 00 00 0 00 + t = {tv} ∈ T and t = {tv } ∈ T such that t − t ∈ k . THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS 15

Then

( 0 00 1 0 00 |tv| + |tv | ≤ 2 · 2 |xv|v = |xv|v if v is archimedean, |tv − tv |v ≤ 0 00 max{tv, tv } ≤ |xv|v if v is nonarchimedean.

0 00 Therefore, a := t − t ∈ 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 ∈ kv with 0 < |x|v ≤ δv, and |xv|v = 1 if δv = 1. We can choose Q 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 Theo- rem 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 1 |a|v < for v ∈ S, and |a|v ≤ for v∈ / S, v 6= v0. δv δv 1 Writing z = a · x and multiplying by a, it is shown that every x ∈ Ak can be expressed as 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 ⊆ + AQ 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 Fr¨ohlich. “Algebraic Number Theory.” London Mathematical Society 2010 [2] Author: Jrgen Neukirch, Translator: Norbert Schappacher. “Algebraic Number Theory.” Springer 1999