Part 1 Basic number theory CHAPTER 1 Algebraic numbers and adèles 1. Valuations of the field of rational numbers We will begin by reviewing the construction of real numbers from ra- tional numbers. Recall that the field of rational numbers Q is a totally ordered field by the semigroup Q of positive rational numbers. We will Å call the function Q£ Q mapping a nonzero rational number x to x or ! Å x depending on whether x Q or x Q the real valuation. It defines ¡ 2 Å ¡ 2 Å a distance on Q with values in Q and thus a topology on Q. A real number is defined as anÅ equivalence class of Cauchy sequences of rational numbers. We recall that a sequence {®i }i N of rational num- 2 bers is Cauchy if for all ² Q , ®i ®j ² for all i, j large enough, and 2 Å j ¡ j Ç two Cauchy sequences are said to be equivalent if in shuffling them we get a new Cauchy sequence. The field of real numbers R constructed in this way is totally ordered by the semigroup R consisting of elements of R which can be represented by Cauchy sequencesÅ with only positive ra- tional numbers. The real valuation can be extended to R£ with range in the semigroup R of positive real numbers. The field of real numbers R is now completeÅ with respect to the real valuation in the sense that ev- ery Cauchy sequences of real numbers is convergent. According to the Bolzano-Weierstrass theorem, every closed interval in R is compact and consequently, R is locally compact. We will now review the construction of p-adic numbers in following the same pattern. For a given prime number p, the p-adic valuation of a nonzero rational number is defined by the formula ordp (m) ordp (n) m/n p¡ Å j jp Æ where m,n Z {0}, and ord (m) and ord (n) are the exponents of the 2 ¡ p p highest power of p dividing m and n respectively. The p-adic valuation is ultrametric in the sense that it satisfies the multiplicative property and the ultrametric inequality: (1.1) ®¯ ® ¯ and ® ¯ max{ ® , ¯ }. j jp Æ j jp j jp j Å jp · j jp j jp The field Qp of p-adic numbers is the completion of Q with respect to the p-adic valuation, its elements are equivalence classes of Cauchy 3 sequences of rational numbers with respect to the p-adic valuation. If {®i }i N is a Cauchy sequence for the p-adic valuation, then { ®i p } is a 2 j j Cauchy sequence for the real valuation and therefore it has a limit in R . This allows us to extend the p-adic valuation to Q as a function . Å: p j jp Qp R that satisfies (1.6). If ® Qp {0} and if ® limi ®i then the ! Å 2 ¡ Æ !1 ultrametric inequality (1.1) implies that ® ® for i large enough. In j jp Æ j i jp particular, the p-adic valuation ranges in pZ, and therefore in Q . A p-adic integer is defined to be a p-adic number of valuationÅ no more than 1, the set of p-adic integers is denoted: (1.2) Z {® Q ® 1}. p Æ 2 p j j jp · LEMMA 1.1. Every p-adic integer can be represented by a Cauchy se- quence made only of integers. PROOF. We can suppose that ® 0 because the statement is fairly ob- 6Æ vious otherwise. Let ® Z {0} be a p-adic integer represented by a 2 p ¡ Cauchy sequence ® p /q where p ,q are relatively prime nonzero i Æ i i i i integers. As discussed above, for large i, we have ® ® 1 so that j jp Æ j i jp · qi is prime to p. It follows that one can find an integer qi0 so that qi qi0 is as i p-adically close to 1 as we like, for example q q0 1 p¡ . The sequence j i i ¡ j · ®0 p q0 , made only of integers, is Cauchy and equivalent to the Cauchy i Æ i i sequence ®i . One can reformulate the above lemma by asserting that the ring of p-adic integers Zp is the completion of Z with respect to the p-adic valu- ation (1.3) Z lim Z/pnZ. p Æ n á¡ It follows that Zp is a local ring, its maximal ideal is pZp and its residue field is the finite field F Z/pZ. p Æ It follows also from (1.3) that Zp is compact. With the definition of Zp as the valuation ring (1.2), it is a neighborhood of 0 in Qp , and in partic- ular, Qp is a locally compact field. This is probably the main property it shares with its cousin R. In constrast with R, the topology on Qp is totally disconnected in the sense every p-adic number ® admits a base of neigh- borhoods made of open and compact subsets of the form ® pnZ with Å p n . ! 1 It is straightforward to derive from the definition of the real and the p-adic valuations that for all ® Q, ® 1 for almost all primes p, and 2 j jp Æ that the product formula Y (1.4) ® ® p 1 j j1 p j j Æ 4 holds. In this formula the product runs the prime numbers, and from now on, we will have the product run over the prime numbers together with the sign , which may be considered as the infinite prime of Q. 1 There are essentially no other valuations of Q other than the ones we already know. We will define a valuation to be a homomorphism . : Q£ R such that the inequality j j ! Å (1.5) ® ¯ ( ® ¯ ) j Å j · j j Å j j is satisfied for all ®,¯ Q£. 2 For all prime numbers p and positive real numbers t, . t is a valua- j jp tion in this sense. For all positive real numbers t 1, . t is also a val- · j j uation. In these two cases, a valuation of the form . t will1 be said to be j jv equivalent to . Equivalent valuations define the same completion of j jv Q. THEOREM 1.2 (Ostrowski). Every valuation of Q is equivalent to either the real valuation or the p-adic valuation for some prime number p. PROOF. A valuation . : Q£ R is said to be nonarchimedean if it is j j ! Å bounded over Z and archimedean otherwise. We claim that a valuation is nonarchimedean if and only if it satisfies the ultrametric inequality (1.6) ® ¯ max{ ® , ¯ }. j Å j · j j j j If (1.6) is satisfied, then for all n N, we have 2 n 1 1 1. j j Æ j Å ¢¢¢ Å j · Conversely, suppose that for some positive real number A, the inequality ® A holds for all ® Z. For ®,¯ Q with ® ¯ , the binomial formula j j · 2 2 j j ¸ j j and (1.5) together imply ® ¯ n A(n 1) ® n j Å j · Å j j for all n N. By taking the n-th root and letting n go to , we get ® ¯ 2 1 j Å j · ® as desired. j j Let . : Q£ R be a nonarchimedean valuation. It follows from the j j ! Å ultrametric inequality (1.6) that ® 1 for all ® Z. The subset p of Z j j · 2 consisting of ® Z such that ® 1 is then an ideal. If ® ¯ 1, then 2 j j Ç j j Æ j j Æ ®¯ 1; in other words ®,¯ p, then ®¯ p. Thus p is a prime ideal of Z, j j Æ ∉ ∉ and is therefore generated by a prime number p. If t is the positive real number such that p p t , then for all ® Q we have ® ® t . j j Æ j jp 2 j j Æ j jp We claim that a valuation . is archimedean if for all integers ¯ 1, we j j È have ¯ 1. We will argue by contradiction. Assume there is an integer j j È 5 ¯ 1 with ¯ 1, then we will derive that for all integers ®, we have È j j · ® 1. One can write j j · ® a a ¯ a ¯r Æ 0 Å 1 Å ¢¢¢ Å r with integers a satisfying 0 a ¯ 1 and a 0. In particular a ¯ i · i · ¡ r È j i j · and r log®/log¯. It follows from (1.5) that · log® ® (1 )¯. j j · Å log¯ Replacing ® by ®k in the above inequality, taking k-th roots on both sides, and letting k tend to , we will get ® 1. 1 j j · We now claim that for every two natural numbers ®,¯ 1 we have È (1.7) ® 1/log® ¯ 1/log¯. j j · j j One can write ® a a ¯ a ¯r Æ 0 Å 1 Å ¢¢¢ Å r with integers a satifying 0 a ¯ 1 and a 0. In particular a ¯ i · i · ¡ r È j i j Ç and r log®/log¯. It follows from (1.5) and ¯ 1 that · j j È µ log®¶ log® ® 1 ¯ ¯ log¯ . j j · Å log¯ j j Replacing ® by ®k in the above inequality, taking k-th roots on both sides, and letting k tend to , we will get (1.7). By symmetry, we then derive the 1 equality (1.8) ® 1/log® ¯ 1/log¯,a j j Æ j j implying . is equivalent to the real valuation . j j j j1 2.