From quadratic reciprocity equation for any integer m is equivalent to solving to Langlands' program the equation in the p-adic integers for all primes p. In addition to the prime solutions the existence of a A major task in mathematics is to solve equa- solution over the real numbers R is of course a neces- tions. Early sources, like the Rhind Papyrus, writ- sary condition in order to have a solution over Z. ten around 1650 BCE by Ahmes, contains methods The p-adic integers is defined as an inverse limit, for solving linear equations. Rational numbers and and has the corresponding "completion" topology. complex numbers were created to solve certain equa- The topology can be defined via the p-adic metric; tions. And still today, equations and their solutions in various number systems, continue to be an excel- Definition 1. For a rational number lent source of new knowledge, in mathematics as well pαm as in other scientific disciplins. q = n Finding integer solutions of equations is of par- ticular interest to mathematicians. This problem is where n and m are not divisible by p, the p-adic met- closely related to the many attempts at understand- ric of q is given by ing the most basic mathematical object, the natu- −α ral numbers; N = f1; 2; 3;::: g. The additive struc- jqjp = p ture of the natural numbers is easily accessible; ev- ery positive integer is obtained from 1 by repeated It follows that two natural numbers are p-close (i.e., addition. The multiplicative structure is more sub- with respect to the p-adic metric) if their difference tle. The prime numbers 2; 3; 5;; 7;::: , which are the is divisible by a high power of p. In this way 14 multiplicative basis for the natural numbers, are still and 15 are not so 2-close since their difference is not hiding a lot of secrets; e.g. how to efficiently decide divisible by any positive power of 2, while 31 and 63 with difference 63 − 31 = 32 = 25 are much 2-closer. whether a given number is a prime or not. ^ A first attempt to decide whether a polynomial We can localize Zp in the multiplicative set of non- ^ zero elements to obtain the p-adic numbers p. equation P (x1; : : : ; xn) = 0 has an integer solution Q is to reduce the problem modulo m, i.e. look for The real numbers R is the completion of the rational numbers with respect to the ordinary norm. We solutions in the ring Zm = f0; 1; 2; : : : ; m − 1g for Q various positive integers m. By using the Chinese use the notation jqj1 for this norm, and call R the ^ Remainder Theorem this is equivalent to the prob- completion of Q at the ”infinite” prime, i.e. R = Q1. ^ lem of finding solutions modulo powers pk, where p Each Qp, p ≤ 1, is called a "local field”, and Q it- is a prime and k ≥ 1. Solving the congruence (this self a "global" field. The Hasse-Minkowski Theorem is the technical term for an equation when we work gives a positive example of the local-global princi- modulo an integer) ple: A statement is valid over Q if and only if it is ^ valid over all local fields Qp for p ≤ 1. k P (x1; : : : ; xn) ≡ 0 (mod p ) Theorem 2 (Hasse-Minkowski). Let Q be a non- is called a "local" problem since we focus on one prime degenerate quadratic form. Then or "place" p 2 Z at a time. The counterpart is a "global" problem, where we solve the equation over Q(x1; : : : ; xn) = 0 the integers Z. Kurt Hensel (1861-1941) reformulated the local has a non-trivial integer solution if and only if it has problem i 1897 by introducing the p-adic integers; a real solution and a p-adic solution for every prime p. ^ = lim =pk Zp − Z Z Notice that the Hasse-Minkowski Theorem tells us k something about quadratic polynomials. The result a construction which embraces all powers pk, for is not true for polynomials of higher degree. It was k ≥ 1. Hensel's reformulation says that solving the already known in 1909 that the Fermat equation xn + yn = zn has p-adic solutions for all p. But as we now The Legendre symbol can be computed by Euler's for- know, there are no non-trivial integer solutions. mula, introduced by Leonhard Euler (1707-1783) in Another example of an equation which is solvable 1748: locally, but not globally was found by Ernst Selmer (1920-2006) in 1951. It is given by the formula Theorem 5. Let p be an odd prime and d an integer. Then we have 3x3 + 4y3 + 5z3 = 0 d p−1 ≡ d 2 (mod p) and Selmer showed that this equation can be solved p modulo p for any prime p, as well as over R, but there An immediate and useful consequence of Euler's are still no solutions in Z. formula is the fact that ( An innocent looking equation which has drawn a −1 1 if p ≡ 1 (mod 4) = lot of attention throughout history is the quadratic p −1 if p ≡ 3 (mod 4) equation x2 = d; Euler's criterion is easily proved using Fermat's little theorem, where d is a positive integer. The equation has an dp−1 ≡ 1 (mod p) integer solution if and only if d is a perfect square. Assume d is not a perfect square. Then by the Rewriting this as Hasse-Minkowski Theorem we know that for some p−1 p−1 prime p the congruence (d 2 − 1)(d 2 + 1) ≡ 0 (mod p) x2 ≡ d (mod pk) and using the fact that p is prime, we deduce that one of the factors has to be congruent to 0 (mod p). fails to have a solution, i.e., for some prime power If d is a quadratic residue, i.e., x2 ≡ d (mod p) for pk the number d fails to be a quadratic residue. some x, then Adrien-Marie Legendre (1752-1833) reformulated p−1 2 p−1 p−1 this statement in a symbolic way: d 2 ≡ (x ) 2 ≡ x ≡ 1 (mod p) Definition 3. Let p be an odd prime and d an inte- which makes the first factor equal to 0. Since the d 2 ger. The Legendre symbol p is defined as polynomial x − d has degree 2, it can have at most two roots, x and −x. Thus there are at least p−1 8 2 1 if x2 ≡ d (mod p) is solvable non-zero quadratic residues. On the other hand the d <> = −1 if x2 ≡ d (mod p) is not solvable polynomial p p−1 :>0 if p divides d x 2 − 1 p−1 can have at most 2 non-zero roots. It follows that The Legendre symbol is multiplicative and p- p−1 periodic in the top argument. It can be shown that the remaining 2 non-quadratic residue classes must the symbols satisfy the quadratic reciprocity law: be roots of the second factor, i.e., satisfying p−1 Theorem 4. Let p and q be two odd primes. Then d 2 ≡ −1 (mod p) p−1 p q (p−1)(q−1) Thus we have d 2 ≡ ±1 (mod p), depending on = (−1) 4 q p whether d is a quadratic residue modulo p or not. The special value for the prime p = 2 is given by A general framework for describing solutions of 2 p2−1 polynomial equations was introduced by Evariste´ = (−1) 8 p Galois (1811-1832) in 1832, a few days before he p p died from wounds suffered in a duel, only 20 years of v + w d 2 Fp[ d]. We have age. Given a polynomial with integer coefficients, we p p can extend the rational numbers by the the solutions (v + w d)p = vp + wp( d)p of the equation (found among complex numbers) to p−1 p = v + wd 2 d form a field extension E of the rational numbers Q. Galois' idea was to consider automorphisms of the ex- p−1 Thus we are interested in the value of d 2 (mod p). tended field fixing the rational numbers, and thereby From Fermat's little theorem we have fixing the polynomial itself. The set of such automor- p p−1 p−1 phisms forms a group, the so-called Galois group 0 = d − d = d · (d 2 − 1)(d 2 + 1) G(E=Q) of the field extension. The Galois groups have a rich structure which reflects a lot of properties and since Fp is a field, one of the three factors must of the original polynomial equation; e.g., the Galois vanish. Our assumption that p does not divide 2d group is trivial if and only if the polynomial splits excludes the first factor, and we are left with the other completely in linear factors with integer coefficients. two. For p−1 The equation x2 −d = 0 has at most two roots, and d 2 = 1; thep only possible non-trivial automorphism ofpE = the Frobenius map is the identity, while for Q( d) fixing Q is the conjugation map σ : d 7! p p−1 − d. Thus the Galois group in this case is the unique d 2 = −1; 2 group of two elements, Z2 = fId; σg where σ = Id. the Frobenius map corresponds to the conjugation The group Z2 also appears in an arithmetic con- p p ∗ automorphism σ : d 7! − d.