P-ADIC NUMBERS JAN-HENDRIK EVERTSE March 2011 Literature: Z.I. Borevich, I.R. Shafarevich, Number Theory, Academic Press, 1966, Chap. 1,4. A. Fr¨ohlich, Local Fields, in: Algebraic Number Theory, edited by J.W.S. Cassels, A. Fr¨ohlich, Academic Press, 1967, Chap. 1. N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap. I,III. C. Lech, A note on recurring sequences, Arkiv f¨ormathematik, Band 2, no. 22 (1953), 417{421. L.J. Mordell, Diophantine Equations, Academic Press, 1969, Chap. 23. 1. Absolute values The p-adic absolute value j·jp on Q is defined as follows: if a 2 Q, a 6= 0 then m −m write a = p b=c where b; c are integers not divisible by p and put jajp = p ; further, put j0jp = 0. −7 8 −3 7 −8 3 Example. Let a = −2 3 5 . Then jaj2 = 2 , jaj3 = 3 , jaj5 = 5 , jajp = 1 for p > 7. We give some properties: jabjp = jajpjbjp for a; b 2 Q; ja + bjp 6 max(jajp; jbjp) for a; b 2 Q (ultrametric inequality): Notice that the last property implies that ja + bjp = max(jajp; jbjp) if jajp 6= jbjp: It is common to write the ordinary absolute value jaj = max(a; −a) on Q as jaj1, to call 1 the `infinite prime' and to define MQ := f1g [ fprimesg. 1 2 MASTER COURSE DIOPHANTINE EQUATIONS, SPRING 2011 Then we have the important product formula: Y jajp = 1 for a 2 Q; a 6= 0: p2MQ Absolute values on fields. We define more generally absolute values on fields. Let K be any field. An absolute value on K is a function j·j : K ! R>0 with the following properties: jabj = jaj · jbj for a; b 2 K; ja + bj 6 jaj + jbj for a; b 2 K (triangle inequality); jaj = 0 () a = 0: Note that these properties imply that j1j = 1. The absolute value j · j is called non-trivial if there is a 2 K with jaj 6= f0; 1g. The absolute value j · j is called non-archimedean if the triangle inequality can be replaced by the stronger ultrametric inequality ja + bj 6 max(jaj; jbj) for a; b 2 K: An absolute value not satisfying the ultrametric inequality is called archimedean. If K is a field with absolute value j · j and L an extension of K, then an extension or continuation of j·j to L is an absolute value on L whose restriction to K is j · j. Examples. 1) The ordinary absolute value j · j on Q is archimedean, while the p-adic absolute values are all non-archimedean. 2) Let K be any field, and K(t) the field of rational functions of K. For a polynomial f 2 K[t] define jfj = 0 if f = 0 and jfj = edeg f if f 6= 0. Further, for a rational function f=g with f; g 2 K[t] define jf=gj = jfj=jgj. Verify that this defines a non-archimedean absolute value on K(t). Two absolute values j · j1; j · j2 on K are called equivalent if there is α > 0 α such that jxj2 = jxj1 for all x 2 K. We state without proof the following result: Theorem 1.1 (Ostrowski). Every non-trivial absolute value on Q is equivalent to either the ordinary absolute value or a p-adic absolute value for some prime number p. P-ADIC NUMBERS 3 Valuations. In algebra and number theory, one quite often deals with val- uations instead of absolute values. A valuation on a field K is a function v : K ! R [ f1g such that for some constant c > 1, c−v(·) defines a non- archimedean absolute value on K. That is, v(x) = 1 () x = 0; v(xy) = v(x) + v(y) for x; y 2 K; v(x + y) > min(v(x); v(y)) for x; y 2 K. The valuation is called non-trivial if there is a 2 K∗ with v(a) 6= 0. The set v(K∗) is an additive subgroup of R. The valuation v is called discrete if v(K∗) is a discrete subgroup of R. A normalized discrete valuation is one for which v(K∗) = Z. 2. Completions An absolute value preserving isomorphism between two fields K1;K2 with absolute values j · j1, j · j2, respectively, is an isomorphism ' : K1 ! K2 such that j'(x)j2 = jxj1 for x 2 K1. 1 Let K be a field, j · j a non-trivial absolute value on K, and fakgk=0 a sequence in K. 1 We say that fakgk=0 converges to α with respect to j · j if limk!1 jak − αj = 0. 1 Further, fakgk=0 is called a Cauchy sequence with respect to j · j if limm;n!1 jam − anj = 0. Notice that any convergent sequence is a Cauchy sequence. We say that K is complete with respect to j · j if every Cauchy sequence w.r.t. j · j in K converges to a limit in K. For instance, R and C are complete w.r.t. the ordinary absolute value. By mimicking the construction of R from Q, one can show that every field K with an absolute value can be extended to an essentially unique field Ke, such that Ke is complete and every element of Ke is the limit of a Cauchy sequence from K. Theorem 2.1. Let K be a field with absolute value j · j. There is an up to absolute value preserving isomorphism unique extension field Ke of K, called the completion of K, having the following properties: 4 MASTER COURSE DIOPHANTINE EQUATIONS, SPRING 2011 (i) j · j can be continued to an absolute value on Ke, also denoted j · j, such that Ke is complete w.r.t. j · j; (ii) K is dense in Ke, i.e., every element of Ke is the limit of a sequence from K. Proof. We give a sketch. Cauchy sequences, limits, etc. are all with respect to j · j. The set of Cauchy sequences in K with respect to j·j is closed under termwise addition and multiplication fang + fbng := fan + bng, fang · fbng := fan · bng. With these operations they form a ring, which we denote by R. It is not difficult to verify that the sequences fang such that an ! 0 with respect to j · j form a maximal ideal in R, which we denote by M. Thus, the quotient R=M is a field, which is our completion Ke. We define the absolute value jαj of α 2 Ke by choosing a representative fang of α, and putting jαj := limn!1 janj, where now the limit is with respect to the ordinary absolute value on R. It is not difficult to verify that this is well-defined, that is, the limit exists and is independent of the choice of the representative fang. We may view K as a subfield of Ke by identifying a 2 K with the element of Ke represented by the constant Cauchy sequence fag. In this manner, the absolute value on Ke constructed above extends that of K, and moreover, every element of Ke is a limit of a sequence from K. So K is dense in Ke. One shows that Ke is complete, that is, any Cauchy sequence fang in Ke has a limit in Ke, by taking very good approximations bn 2 K of an and then taking the limit of the bn. Finally, if K0 is another complete field with absolute value extending that on K such that K is dense in K0 one obtains an isomorphism from Ke to K0 as follows: Take α 2 Ke. Choose a sequence fakg in K converging to α; this is 0 necessarily a Cauchy sequence. Then map α to the limit of fakg in K . 2 Corollary 2.2. Assume that j · j is a non-archimedean absolute value on K. Then the extension of j · j to Ke is also non-archimedean. Proof. Let a; b 2 Ke. Choose sequences fakg, fbkg in K that converge to a, b, respectively. Then taking the limit of jak + bkj 6 max(jakj; jbkj) gives ja + bj 6 max(jaj; jbj). 2 P-ADIC NUMBERS 5 Ostrowski proved that any field complete with respect to an archimedean absolute value is isomorphic to R or C. As a consequence, any field that can be endowed with an archimedean absolute value is isomorphic to a subfield of C. On the other hand, there is a much larger variety of fields with a non- archimedean absolute value. It is possible to define notions such as convergence, continuity, differentia- bility, etc. for complete fields with a non-archimedean absolute value similarly as for R or C, and this leads to non-archimedean analysis. One of the striking features of non-archimedean analysis is the following very easy criterion for convergence of series. Lemma 2.3. Let K be a field complete w.r.t. a non-archimedean absolute 1 P1 value j · j. Let fakgk=0 be a sequence in K. Then k=0 ak converges in K if and only if limk!1 ak = 0. P1 Proof. Suppose that α := k=0 ak converges. Then n n−1 X X an = ak − ak ! α − α = 0: k=0 k=0 Pn Conversely, suppose that ak ! 0 as k ! 1. Let αn := k=0 ak. Then for any integers m; n with 0 < m < n we have n X jαn − αmj = j akj 6 max(jam+1j;:::; janj) ! 0 as m; n ! 1 : k=m+1 So the partial sums αn form a Cauchy sequence, hence must converge to a limit in K.