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The P-Adic Numbers and Basic Theory of Valuations

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The P-Adic Numbers and Basic Theory of Valuations 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 as- sume 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 j · j : k ! R≥0 satisfying these axioms: (1) jαj = 0 () α = 0 (2) jαβj = jαjjβj (3) There is a constant C ≥ 1 such that j1 + αj < C whenever jαj ≤ 1. 1 2 ANDREW DING Definition 1.2. The trivial valuation on k is the following: ( 0 if α = 0; jαj = 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 j1j = j1j · j1j =) j1j = 1: Further, given ! 2 k, if there exists n 2 N such that !n = 1, then j!j = 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 j · j; j · j0 on Q are equivalent if there exists some constant c > 0 such that jxj0 = jxjc: Note that if jxj is a valuation, then jxjc 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 2 k jα + βj ≤ jαj + jβj 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 j · j on k is called discrete if there exists a δ > 0 such that if α 2 k satisfies 1 − δ < jαj < 1 + δ; then jαj = 1. One equivalent condition is that the group consisting of ln α is a discrete (addi- tive) subgroup of R. Such a group is necessarily isomorphic to Z. Definition 2.2. Given a discrete valuation j · j on k, the value group is the (multiplicative) group S := fjαj 2 R≥0 : α 2 kg ⊆ R≥0: From the above, it is clear that S = fcm : m 2 Zg for some 0 < c < 1. If jαj = cm, then we call m the order of α. THE P-ADIC NUMBERS AND BASIC THEORY OF VALUATIONS 3 Definition 2.3. A valuation j · j 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 2 k, jα + βj ≤ maxfjαj; jβjg: 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 infi- nite. We make the following remark for future use. Remark 2.5. If jαj < jβj, then jα + βj = jβj. This is because jαj = j(α + β) − βj ≤ maxfjα + βj; jβjg: Definition 2.6. Suppose j · j is a nonarchimedean valuation on k. The set o := fα 2 k : jαj ≤ 1g is a ring, called the valuation ring. It has a unique maximal ideal p := fα 2 k : jαj < 1g: We have two statements we can say about valuations. Lemma 2.7. A nonarchimedean valuation j · j is discrete if and only if p is a principal ideal. Lemma 2.8. A necessary and sufficient condition for a valuation j · j on k to be nonarchimedean is that jnj ≤ 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 j · j is an nonarchimedean valuation. For any n, we have jnj = j1 + ··· + 1j ≤ j1j = 1: Conversely, suppose jnj ≤ 1 for all n in the subring generated by 1. Assume WLOG that jxj ≥ jyj. Using the binomial theorem, n X n jx + yjn = jxjn−kjyjk ≤ (n + 1)Njxjn: k k=0 Therefore, jx + yj ≤ ((n + 1)N)1=njxj = ((n + 1)N)1=n maxfjxj; jyjg: By letting n ! 1, we obtain the desired inequality. Corollary 2.9. If char k = p (p nonzero), then all valuations on k are nonar- chimedean. Proof. The ring generated by 1 in k is a finite field. Every element α satisfies jαj = 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 nonar- chimedean 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) 2 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) 2 k0[t] and p(t) - u(t) and p(t) - v(t). In addition, there is the non-archimedean valuation j · j1 given by u(t) deg v−deg u (3.3) = c : v(t) 1 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) = fβ 2 k : jβ − αj < dg; where α 2 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 j · j; j · j0 on the same field k induce the same topology. Then they are equivalent. n Proof. Let α 2 k. We have jαj < 1 () limn!+1 α = 0. It follows that jαj < 1 () jαj0 < 1. Taking inverses, we have jαj > 1 () jαj0 > 1. By elimination, we can conclude that jαj = 1 () jαj0 = 1. Let β; γ 2 k be nonzero, and let m; n 2 Z. By letting α = βmγn, we have m ln jβj + n ln jγj >; <; or = 0 () m ln jβj0 + n ln jγj0 >; <; or = 0; respectively. It follows that ln jβj ln jγj = ln jβj0 ln jγj0 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 j · j1;:::; j · jn be inequivalent nontrivial valuations over a field k. Then given any αj 2 k(1 ≤ j ≤ n), and > 0, there exists a ξ 2 k that satisfies jαi − ξji < simultaneously. The Weak Approximation Theorem is related to the Chinese Remainder The- orem. If we let k = Q, and have n p-adic valuations j · jpi , for any n-tuple of rational numbers (a1; : : : ; an), we can find a x 2 Q such that jx − aijpi < , that is, k x − ai ≡ 0 mod pi for arbitrary k 2 Z.
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