Geometry of p-adic numbers Zair Ibragimov CSUF Valuations on Geometry of p-adic numbers Rational Numbers Completions of Rational Numbers Zair Ibragimov The p-adic CSUF numbers Geometry of p-adic integers Talk at Fullerton College July 14, 2011 Valuations on Rational Numbers Geometry of p-adic Let = f:::; −2; −1; 0; 1; 2;::: g denote the integers. Let numbers Z = fa=b : a; b 2 and b > 0g denote the rational numbers. Zair Ibragimov Q Z CSUF Valuations on We can add and multiply rational numbers: Rational Numbers a=b + c=d = (ad + bc)=(bd) and (a=b) · (c=d) = (ac)=(bd) Completions of Rational Numbers By a valuation on Q we mean a function v : Q ! [0; +1) The p-adic satisfying the following three properties: numbers Geometry of 1 v(x) ≥ 0, and v(x) = 0 if and only if x = 0; p-adic integers 2 v(x + y) ≤ v(x) + v(y) (the triangle inequality); 3 v(xy) = v(x)v(y). We say that a valuation v is non-Archimedean if it satisfies the strong triangle inequality: v(x + y) ≤ maxfv(x); v(y)g. The standard valuation on Q Geometry of p-adic The real absolute value function j · j on is a well-known numbers Q example of a valuation. Zair Ibragimov CSUF Recall that jxj = x if x ≥ 0 and jxj = −x if x < 0. Valuations on Rational Properties (1) and (3) are easily verified. Numbers Completions To show property (2), enough to consider three cases: of Rational Numbers Case 1: x; y ≥ 0. Then jx + yj = x + y = jxj + jyj. The p-adic numbers Case 2: x; y ≤ 0. Then jx + yj = −x − y = jxj + jyj. Geometry of p-adic integers Case 3: x ≤ 0 and y ≥ 0. Then jx + yj is either x + y or −x − y and jxj + jyj = −x + y. Hence jx + yj ≤ jxj + jyj. The function j · j0 (defined by jxj0 = 0 if x = 0 and jxj0 = 1 if x 6= 0) is easily seen to be a valuation on Q. It is referred to as the trivial valuation on Q. Fundamental Theorem of Arithmetics Geometry of p-adic numbers Theorem (Fundamental Theorem of Arithmetics) Zair Ibragimov CSUF Any integer can be written as a product of primes. That is, Valuations on a a a Rational jnj = p 1 · p 2 ··· p k ; Numbers 1 2 k Completions of Rational where p1; p2;:::; pk are primes and a1; a2;:::; ak are positive Numbers integers. The p-adic numbers For example, 360 = 22 · 32 · 5 and 93555 = 34 · 52 · 7 · 11. Geometry of p-adic integers If we allow a1; a2;:::; ak to be any integers (positive or negative), then we can write any rational number as a product a1 a2 ak of primes. That is, jxj = p1 · p2 ··· pk . 360 2 −2 −1 −1 −1 For example, 93555 = 2 · 3 · 5 · 7 · 11 . The p-adic valuation on Q Geometry of Let p be a fixed prime number, i.e., p = 2; 3; 5; 7; 11; 13;::: . p-adic numbers Any non-zero rational number x can be written uniquely as Zair Ibragimov a CSUF x = pn ; b Valuations on Rational where neither a not b is divisible by p and n 2 Z. The p-adic Numbers absolute value on is defined by Completions Q of Rational ( Numbers 0 if x = 0 jxjp = The p-adic −n numbers p if x 6= 0: Geometry of 2 −2 −1 −1 −1 p-adic integers For example, since 360=93555 = 2 · 3 · 5 · 7 · 11 , −2 2 j360=93555j2 = 2 , j360=93555j3 = 3 , j360=93555j7 = 7 and 0 j360=93555jp = p = 1 for all p 6= 2; 3; 5; 7; 11. Problem Show that the p-adic absolute value is a non-Archimedean valuation on Q. Show that jxjp ≤ 1 if and only if x 2 Z. Ostrowski's Theorem Geometry of p-adic Two valuations v1 and v2 on Q is said to be equivalent if there numbers exists c > 0 such that Zair Ibragimov CSUF c v2(x) = v1(x) for all x 2 : Valuations on Q Rational Numbers Completions Theorem (Ostrowski) of Rational Numbers Any valuation on Q is equivalent to either the trivial valuation The p-adic numbers or the real absolute value j · j or a p-adic absolute value j · jp for Geometry of some prime p. p-adic integers The classical Analysis, Algebra and Geometry is based on the standard valuation on Q. The p-adic valuation, which is the topic of my talk, has been introduced in 1897 by Kurt Hensel. A systematic study of p-adic numbers began in the late 1960's. Metrics and Ultrametrics on Q Geometry of p-adic A distance function d : Q × Q ! [0; +1) is called a metric on numbers if (1) d(x; y) ≥ 0 and d(x; y) = 0 if and only if x = y, Zair Ibragimov Q CSUF (2) d(x; y) = d(y; x) and (3) d(x; y) ≤ d(x; z) + d(z; y). Valuations on It is called an ultrametric if it satisfies (1), (2) and Rational Numbers (3'): d(x; y) ≤ maxfd(x; z); d(z; y)g. Completions of Rational Each valuation v on Q induces a metric d to measure a Numbers distance between points in Q by d(x; y) = v(x − y). The p-adic numbers The real absolute value j · j induces Euclidean metric on Q, Geometry of p-adic integers given by d(x; y) = jx − yj. As we mentioned above, much of classical analysis and geometry is based on Euclidean metric. Problem Show that the distance function d(x; y) = jx − yjp on Q induced by the p-adic absolute value j · jp is an ultrametric. Completions of Q Geometry of The rational numbers , equipped with Euclidean metric or p-adic Q numbers p-adic ultrametric, is not adequate to study problems in Zair Ibragimov CSUF Algebra or Analysis. 2 Valuations on For example, the polynomial equation x − 2 = 0 in Q is not Rational p Numbers solvable in Q because 2 is not a rational number. Completions of Rational Similarly, the series Numbers 1 1 The p-adic X numbers k! k=0 Geometry of p-adic integers in Q does not converge in Q because its sum is equal to e, which is not a rational number. Overcoming these problems require a completion of Q. In other words, we need to add more points to Q. Completions of Q depend on the metric used and is based on the notion of Cauchy sequences. Completion of Q with respect to Euclidean metric Geometry of We say that a sequence fx g of numbers in is a Cauchy p-adic n Q numbers sequence if jxn − xmj ! 0 as n; m ! 1. For example, a trivial Zair Ibragimov CSUF sequence x; x;::: is a Cauchy sequence. So is the sequence 1; 1=2; 1=3;::: . Valuations on Rational Numbers Two Cauchy sequences fxng and fyng are said to be equivalent Completions if jxn − ynj ! 0 as n ! 1. Let [fxng] denote the set of all of Rational Numbers Cauchy sequences in Q that are equivalent to fxng. The p-adic numbers The set of all equivalent Cauchy sequences, equipped with the Geometry of metric d([fx g]; [fy g]) = lim jx − y j, is called the p-adic integers n n n!1 n n metric completion of Q. This space can be identified with the real numbers R. Problem p Show that a sequence converging to 2 is a Cauchy sequence. Show that all such sequences are equivalent. Completion of Q with respect to p-adic ultrametric Geometry of p-adic The p-adic numbers, denoted by p, is obtained as the numbers Q completion of the rational numbers with respect to the Zair Ibragimov Q CSUF p-adic ultrametric. Valuations on Rational Since every rational numbers x can be identified with a Cauchy Numbers sequence fx; x;::: g, every rational number is a p-adic number. Completions of Rational That is, Q ⊂ Qp. But there are much more p-adic numbers Numbers that cannot be identified with rational numbers. In fact, while The p-adic numbers the rational numbers Q are countable, the p-adic numbers Qp Geometry of are uncountable. p-adic integers There are significant differences between the two completions of Q due to the fact that Euclidean metric satisfies only the triangle inequality while the p-adic metric satisfies the strong triangle inequality. For example, R is a connected space while Qp is totally disconnected like the middle-third Cantor set. Completion of Z with respect to p-adic ultrametric Geometry of p-adic numbers The integers Z, equipped with Euclidean metric jn − mj, is Zair Ibragimov complete. This is because there are no Cauchy sequences in CSUF Z except for trivial ones like f1; 1; 1;::: g, f2; 2; 2;::: g and so on. Valuations on Rational So the completion of Z with respect to Euclidean metric is the Numbers same as Z. That is, this completion does not add any new Completions of Rational points to Z. Numbers The p-adic On the other hand, the same integers Z, equipped with the numbers p-adic metric jn − mjp, is NOT complete! For example, the Geometry of p-adic integers following sequence of integers is a nontrivial Cauchy sequence 1; 1 + p; 1 + p + p2; 1 + p + p2 + p3;::: The p-adic integers, denoted by Zp, is the completion of Z with respect to the p-adic ultrametric.