p-adic numbers, LTCC 2010 Manuel Breuning King's College London Contents Notation 1 1. Absolute values and completion 2 1.1. Absolute values 2 1.2. The topology of a valued ¯eld 2 1.3. Absolute values on the rational numbers 3 1.4. Completion 4 1.5. Archimedean absolute values 6 2. The ¯elds Qp and Cp 6 2.1. Algebraic extensions of a complete ¯eld 6 2.2. The ¯eld Cp 7 2.3. Sequences and series 7 2.4. The residue class ¯eld 8 2.5. The value group 9 2.6. The ring Zp 10 2.7. Topology of Qp and Cp 11 3. Normed spaces 11 3.1. Basic de¯nitions 12 3.2. Bounded linear maps 12 3.3. Examples 14 4. Continuous functions on Zp 14 4.1. Mahler expansion: statement of main result 14 4.2. Mahler expansion: proof 15 4.3. The function x 7! ax 18 4.4. The p-adic gamma function 18 5. Di®erentiation 20 5.1. Di®erentiability and strict di®erentiability 20 5.2. Local invertibility of strictly di®erentiable functions 22 5.3. Further results on strictly di®erentiable functions 23 References 23 Notation In this course we will use the following standard notations: N = f1; 2; 3;::: g; Z = f:::; ¡2; ¡1; 0; 1; 2;::: g; Q = rational numbers; R = real numbers; C = complex numbers: Furthermore we will need the following sets: Z¸0 = f0; 1; 2;::: g; R>0 = positive real numbers; R¸0 = non-negative real numbers: 1 2 p-adic numbers, LTCC 2010 By a ring we will always mean a commutative ring with 1. If R is a ring then R£ denotes the group of units in R, so in particular if R is a ¯eld then R£ = R n f0g. 1. Absolute values and completion In this chapter we de¯ne absolute values on ¯elds, construct all absolute values on the ¯eld of rational numbers Q, and discuss the completion of a valued ¯eld. 1.1. Absolute values. We write R¸0 for the set of non-negative real numbers. De¯nition 1.1. Let K be a ¯eld. An absolute value on K is a function j j : K ! R¸0 that satis¯es the following conditions. (1) jxj = 0 if and only if x = 0 (2) jxyj = jxj ¢ jyj for all x; y 2 K (3) jx + yj · jxj + jyj for all x; y 2 K We say that an absolute value j j on K is non-archimedean if it satis¯es the following strengthening of (3). (3') jx + yj · maxfjxj; jyjg for all x; y 2 K We say that an absolute value j j on K is archimedean if it is not non-archimedean. Some authors use the word norm or valuation instead of absolute value. A pair (K; j j) consisting of a ¯eld K and an absolute value j j on K is called a valued ¯eld. We will sometimes refer to K as a valued ¯eld if the absolute value j j is clear from the context. If j j is an absolute value on K then one easily sees that j1j = 1 and j¡xj = jxj for all x 2 K. Example 1.2. Let K be any ¯eld and de¯ne j j : K ! R¸0 by ( 0 if x = 0; jxj = 1 if x 6= 0: One easily sees that this de¯nes a non-archimedean absolute value on K. It is called the trivial absolute value on K. Remark 1.3. If K is a ¯eld and n 2 N then we also write n for the element 1+1+¢ ¢ ¢+1 2 K (where 1+1+¢ ¢ ¢+1 has n summands, and 1 is the multiplicative identity of K). Now if j j is a non-archimedean absolute value on K, then for every n 2 N we have jnj · 1 (this follows by induction using jnj = j(n ¡ 1) + 1j · maxfjn ¡ 1j; j1jg). One can show that the converse of this statement is also true, i.e. if an absolute value j j on K has the property that jnj · 1 for all n 2 N then j j is non-archimedean; for a proof see e.g. [Schikhof, x8]. p 2 2 Example 1.4. The usual absolute value jx + iyjC = x + y is an absolute value on the ¯eld of complex numbers C. This absolute value is archimedean (because since j2jC = 2 > 1 it is not non-archimedean by the previous remark). Exercise 1.5. Let K be a ¯nite ¯eld. Show that K has no non-trivial absolute values. 1.2. The topology of a valued ¯eld. Let (K; j j) be a valued ¯eld. Lemma 1.6. The function d : K £ K ! R¸0 de¯ned by d(x; y) = jx ¡ yj is a metric on K. We call d the metric induced by the absolute value j j. Proof. Clear. ¤ p-adic numbers, LTCC 2010 3 If j j is a non-archimedean absolute value, then the induced metric satis¯es the ultrametric inequality d(x; z) · maxfd(x; y); d(y; z)g for all x; y; z 2 K: De¯nition 1.7. Let x 2 K and " > 0. (1) The set B<"(x) = fy 2 K : jy ¡ xj < "g is called the open ball with radius " and centre x. (2) The set B·"(x) = fy 2 K : jy ¡ xj · "g is called the closed ball with radius " and centre x. From the metric d we obtain a topology on K which we call the topology induced by j j. The set of all open balls B<"(x) (with x 2 K and " > 0) is a basis of this topology. So a valued ¯eld (K; j j) has a natural metric and topology, therefore it makes sense to talk about open sets in K, limits of sequences in K, continuous functions K ! K, etc. Proposition 1.8. K is a topological ¯eld, i.e. the ¯eld operations (1) K £ K ! K, (x; y) 7! x + y (2) K £ K ! K, (x; y) 7! xy (3) K ! K, x 7! ¡x (4) K n f0g ! K n f0g, x 7! x¡1 are continuous. Proof. Let's prove statement (2) in detail. We must show that if (x; y) 2 K £K and " > 0 then there exists a ± > 0 such that the open neighbourhood B<±(x) £ B<±(y) of (x; y) in K £ K is mapped into B<"(xy) under the multiplication map. Now if (u; v) 2 B<±(x) £ B<±(y) then juv ¡ xyj = j(u ¡ x)(v ¡ y) + (u ¡ x)y + (v ¡ y)xj · ju ¡ xj ¢ jv ¡ yj + ju ¡ xj ¢ jyj + jv ¡ yj ¢ jxj · ± ¢ ± + ± ¢ jyj + ± ¢ jxj: So for su±ciently small ± > 0 (where su±ciently small depends on jxj and jyj but not on u and v) we have juv ¡ xyj < ", i.e. uv 2 B<"(xy) as required. The proofs of statements (1),(3) and (4) are similar. ¤ Exercise 1.9. Show that the function K ! R¸0, x 7! jxj is continuous. 1.3. Absolute values on the rational numbers. We now consider absolute val- ues on the ¯eld of rational numbers Q. We will denote the usual absolute value on Q by j j1, so ( x if x ¸ 0; jxj1 = ¡x if x < 0: Clearly j j1 is an archimedean absolute value on Q. Now ¯x a prime number p. We will de¯ne a non-archimedean absolute value j jp on Q, the p-adic absolute value. First let x 2 Q£. By the fundamental theorem e e1 er of arithmetic we can write x = §p q1 ¢ ¢ ¢ qr where q1; : : : ; qr are non-zero prime ¡e numbers di®erent from p and e; e1; : : : ; er 2 Z. We de¯ne jxjp = p . For x = 0 we de¯ne j0jp = 0. Lemma 1.10. j jp is a non-archimedean absolute value on Q. Proof. Conditions (1) and (2) of an absolute value are clearly satis¯ed. It remains to prove condition (3'). We ¯rst observe that if x = 0 or y = 0 or x + y = 0 then 4 p-adic numbers, LTCC 2010 condition (3') is clearly true, so we can assume that x; y; x + y 2 Q£. We write e e1 er f f1 fr g g1 gr x = §p q1 ¢ ¢ ¢ qr , y = §p q1 ¢ ¢ ¢ qr and x + y = §p q1 ¢ ¢ ¢ qr . Then minfe;fg minfe1;f1g minfer ;fr g 0 x = p q1 ¢ ¢ ¢ qr ¢ x ; minfe;fg minfe1;f1g minfer ;fr g 0 y = p q1 ¢ ¢ ¢ qr ¢ y for some x0; y0 2 Z n f0g. It follows that minfe;fg minfe1;f1g minfer ;fr g 0 0 x + y = p q1 ¢ ¢ ¢ qr ¢ (x + y ); hence g g1 gr minfe;fg minfe1;f1g minfer ;fr g p q1 ¢ ¢ ¢ qr = p q1 ¢ ¢ ¢ qr ¢ z for some z 2 Z n f0g. From this we can deduce that g ¸ minfe; fg. It follows that ¡g ¡ minfe;fg jx + yjp = p · p = pmaxf¡e;¡fg ¡e ¡f = maxfp ; p g = maxfjxjp; jyjpg: This completes the proof of condition (3'). ¤ Hence for each prime number p we obtain a non-archimedean absolute value j jp on Q. Furthermore we have the archimedean absolute value j j1 on Q and the trivial absolute value. Theorem 1.12 below shows that this is essentially the complete list of absolute values on Q. De¯nition 1.11. Two absolute values on a ¯eld K are called equivalent if they induce the same topology on K. One can show that two absolute values j j and k k on K are equivalent if and only if there exists a positive real number ® such that jxj = kxk® for all x 2 K.