ÐÏࡱá>þÿ 7îð)þÿÿÿêëìí_û€Cì€A’“”•–~  €  ‚ ƒ „ » Œ œ Ÿ ¿ À Á €€‹ŒŽ‘€Çr€€Ût€ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿì¥Á5@ ø¿0Ÿ·bjbjÏ2Ï2 2­X­X?ÿÿÿÿÿÿˆÚÚÚÚÚÚÚîZuZuZu8’uÌ^}Dîˆôì®~ 0 be integers. Then  EMBED Equation.3 if and only if  EMBED Equation.3  Proof:  EMBED Equation.3  If EMBED Equation.3 , then by definition a and b have the same remainder when divided by n, so by the division algorithm a = nk + r and b = nm + r for some integers k, m, and r with EMBED Equation.3 . Now if we solve both of these equations for r we have  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  If  EMBED Equation.3 then  EMBED Equation.3  which is what we wanted to show.  EMBED Equation.3  Now assume that EMBED Equation.3 . By the division algorithm there exists integers k and m such that, a = nk +  EMBED Equation.3  and b = nm +  EMBED Equation.3 , where  EMBED Equation.3  and EMBED Equation.3 . Subtracting b from a we obtain the equation EMBED Equation.3 , and since it was assumed  EMBED Equation.3 , it must be that  EMBED Equation.3 . But  EMBED Equation.3  and so  EMBED Equation.3  and the only multiple of n in this interval is 0. Thus we have that  EMBED Equation.3  or equivalently  EMBED Equation.3 , and so  EMBED Equation.3  which is what we wanted to show. ²% The reader may think that we do not need a new and slightly longer way to say EMBED Equation.3 . However the notation EMBED Equation.3 is useful because it suggests an equation, and in fact congruences share many properties with equations. The next couple of theorems list a number of ways in which congruences behave like equations. Theorem 1: (Basic Properties for Congruences: Let p > 0 be a fixed integer and a, b, c, and d be arbitrary integers. Then the following properties hold: (1)  EMBED Equation.3 . (2) If  EMBED Equation.3 , then  EMBED Equation.3 . (3) If  EMBED Equation.3  and  EMBED Equation.3 , then  EMBED Equation.3 . (4) If  EMBED Equation.3 and  EMBED Equation.3 , then  EMBED Equation.3  and  EMBED Equation.3 . (5) If  EMBED Equation.3 , then  EMBED Equation.3 and  EMBED Equation.3 . (6) If  EMBED Equation.3 , then  EMBED Equation.3 for any positive integer k. Theorem 2: (Cancellation Law): If a is relatively prime to p and x and y are integers such that  EMBED Equation.3 , then EMBED Equation.3  Proof: By definition  EMBED Equation.3  means p divides  EMBED Equation.3 . Given that a and p are relatively prime p must divide (x – y) since the prime factorization is unique by the Fundamental Theorem of Arithmetic. If p divides (x – y) then by definition EMBED Equation.3  which is what we wanted to show. ²% Definition 2: (Least Residue): If n > 0 and r is the remainder when the Division Algorithm is used to divide b by n, then r is called the least residue of b modulo n. Examples: The least residue of 13 modulo 5 is 3. The least residue of 5 modulo 13 is 5. The least residue of -13 modulo 5 is 2. The least residue of  2 modulo 3 is 1. ²% Lemma 2: If a is relatively prime to p, p > 0, then there exists a unique integer x modulo n such that  EMBED Equation.3  Proof: Let (a, n) = 1 then by the Euclidean Algorithm we can write 1 as a linear combination of a and n, that is,  EMBED Equation.3 for some integers x and y. Rearranging the terms in this equation we have  EMBED Equation.3 . Hence  EMBED Equation.3 which implies that  EMBED Equation.3 by the definition of congruence. To prove uniqueness, suppose that  EMBED Equation.3 and  EMBED Equation.3 . Then  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  (since (a, n) = 1) By definition of congruence if  EMBED Equation.3  then  EMBED Equation.3 . Therefore the solution x must be unique modulo n. ²% Definition 3: Let a be any integer. Any integer x for which  EMBED Equation.3 is called the inverse of a. Definition 4: Let n be a positive integer, and let a be an integer such that EMBED Equation.3  Then a is called a quadratic residue modulo n if the congruence  EMBED Equation.3  EMBED Equation.3  is solvable, and a quadratic nonresidue otherwise. Definition 5: (The Legendre Symbol): Suppose p is an odd prime not dividing a. We define the symbol  EMBED Equation.3  to be 1 or – 1 according to whether a is a quadratic residue or quadratic nonresidue mod p. This is called the Legendre Symbol, after the French mathematician Adrien Marie Legendre (1752 – 1833). We also call  EMBED Equation.3  the quadratic character of a with respect to p. If  EMBED Equation.3  then a is a quadratic residue modulo p which means the congruence EMBED Equation.3  is solvable. In fact, if a is a quadratic residue modulo p then the congruence has exactly two solutions modulo p. This fact is verified through the following theorem and proposition. Theorem 3: (Lagrange’s Theorem): If p is a prime and  EMBED Equation.3   EMBED Equation.3  is a polynomial of degree  EMBED Equation.3  with integral coefficients, then the congruence  EMBED Equation.3  has at most n incongruent solutions modulo p. Proof: See Burton page 191. ²% Proposition 1: Let p be an odd prime and a be any integer such that  EMBED Equation.3  If the congruence  EMBED Equation.3 is solvable, then there are exactly two incongruent solutions modulo p. Proof: Suppose that the congruence  EMBED Equation.3 is solvable and let  EMBED Equation.3  be one of the solutions. Note  EMBED Equation.3  since  EMBED Equation.3  Then there must also be a second solution  EMBED Equation.3  since  EMBED Equation.3  and  EMBED Equation.3 . This second solution is not congruent to the first. To see this, note that  EMBED Equation.3  implies that  EMBED Equation.3  or equivalently EMBED Equation.3  which is impossible. Now by Lagrange’s Theorem, these two solutions exhaust the incongruent solutions of  EMBED Equation.3 . Thus, if  EMBED Equation.3 is solvable there are exactly two incongruent solutions modulo p. ²% For the remainder of this paper we will always assume that p is an odd prime such that a is not divisible by p. Next we will look at an example that incorporates all of the previous definitions. Example 1: When a = 3 and p = 11, the quadratic character of 3 with respect to 11 is equal to 1, or  EMBED Equation.3 , because there exists an x such that  EMBED Equation.3 is solvable. To find x we simply square the numbers from 1 to 10, subtract 3, and see if the result is divisible by 11. Thus we have that  EMBED Equation.3  not divisible by 11  EMBED Equation.3  not divisible by 11  EMBED Equation.3  not divisible by 11  EMBED Equation.3  not divisible by 11  EMBED Equation.3  divisible by 11. Thus,  EMBED Equation.3 is a solution.  EMBED Equation.3  divisible by 11. So,  EMBED Equation.3 is also a solution. Recall by Proposition 1 that if a is a quadratic residue modulo p, then the congruence has exactly two solutions. Thus, we can stop here. We should also note,  EMBED Equation.3 . So we could write that the solutions to the congruence  EMBED Equation.3  are exactly EMBED Equation.3 . It is easy to see that this fact is true since,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  is clearly true. A similar argument would show that the congruence holds for EMBED Equation.3 . Now it is a little deceptive to say that there are only two solutions to the congruence because actually there are infinitely many solutions. It just happens that any other solution will be congruent to  EMBED Equation.3 . For example, 16 is also a solution to the congruence since  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  This statement is true because 253 = (23)(11). Similarly 27, 38, 49, . . . . . , 5 + 11k where k is any integer, are all solutions to the congruence  EMBED Equation.3 . The integers -16, -27, -38, . . . . . . . , - 5 – 11k where k is any integer are also solutions because each of these integers are congruent to  EMBED Equation.3 . ²% The quadratic residues for a system can be found by squaring the numbers from 1 to p  1 and reducing them modulo p as we will see in the following example. Example 2: When p = 11, the quadratic residues can be found by squaring the numbers from 1 to 10 and reducing them modulo 11. Squaring the numbers from 1 to 5 yields:  EMBED Equation.3  Squaring the numbers from 6 to 10 yields the same residues in reverse order:  EMBED Equation.3  So the quadratic residues mod 11 are 1, 3, 4, 5, 9 and the quadratic nonresidues are 2, 6, 7, 8, 10. ²% Since the numbers from 1 to  EMBED Equation.3  yield the same residues as the numbers from  EMBED Equation.3  to p, the quadratic residues can be found simply by squaring the numbers from 1 to  EMBED Equation.3 . Theorem 4: Let p be an odd prime. Then half of the numbers between 1 and p – 1 are quadratic residues of p, and half are nonresidues. Proof: Any quadratic residue of p must be congruent modulo p to one of the numbers  EMBED Equation.3 as was illustrated in the example above. Note that  EMBED Equation.3  So the squares of the numbers from  EMBED Equation.3  up to p – 1 are congruent to the squares of the numbers  EMBED Equation.3  down to 1. All we have left to show is that each of the squares of  EMBED Equation.3  is not congruent to any one of the other squares. We will do this by contradiction. Suppose that there exist two squares such that  EMBED Equation.3  where i and j are distinct and both in the range 1 to  EMBED Equation.3  Then  EMBED Equation.3   EMBED Equation.3  or  EMBED Equation.3  These last two statements are always false because both i and j lie within the range  EMBED Equation.3  Thus, i – j and i + j are both too small to be divisible by p. So we have shown that none of the squares are congruent mod p. Thus, there must be exactly  EMBED Equation.3  quadratic residues of p. ²% In general, the above theorem says that there are exactly  EMBED Equation.3  quadratic residues and exactly  EMBED Equation.3  nonresidues of p. The last theorem in this section, Euler’s Criterion, will be used throughout the discussion of both proofs of the Law of Quadratic Reciprocity presented in the sections Geometric Proof and Schmidt’s Proof below. In order to appropriately discuss the proof of Euler’s Criterion we must first look at a preceding theorem known as Fermat’s Little Theorem. Theorem 5: (Fermat’s Little Theorem): Let p be any prime number. If a is any integer such that  EMBED Equation.3 then  EMBED Equation.3  Proof: Consider the set P = {1a, 2a, 3a, . . . . . . . , (p – 1)a}. By the Cancellation Law each of these numbers must be distinct modulo p. So modulo p, the set P is the same as the set of integers N = {1, 2, 3, . . . . . . . , (p – 1)}. So if we multiply the elements of these two sets together, we would get the same result modulo p, that is,  EMBED Equation.3  Regrouping the left hand side we obtain:  EMBED Equation.3  or equivalently  EMBED Equation.3  To finish the proof we need only to cancel the common term of (p – 1)! from both sides. Given that p is prime, p and (p – 1)! can have no factors in common. Thus by the previous lemma we can divide both sides by (p – 1)! yielding  EMBED Equation.3  which is what we wanted to show. ²% Theorem 6: (Euler s Criterion): Let p be an odd prime and a any integer such that EMBED Equation.3 . Then a is a quadratic residue modulo p if and only if  EMBED Equation.3  Proof:  EMBED Equation.3 Let a be a quadratic residue modulo p. Then there must exist an x such that  EMBED Equation.3 . Since it was assumed that  EMBED Equation.3  this implies  EMBED Equation.3  Thus by Fermat’s Little Theorem we have  EMBED Equation.3  Hence if a is a quadratic residue modulo p then  EMBED Equation.3 which is what we wanted to show. Recall that if (r, p) = 1 and r has order EMBED Equation.3 modulo p (where EMBED Equation.3 is the largest possible order), then r is called a primitive root of p.  EMBED Equation.3  Now suppose that the congruence  EMBED Equation.3 holds and let r be a primitive root of p, r exists because p is assumed to prime. If r is a primitive root of p then  EMBED Equation.3  for some integer m, with  EMBED Equation.3  Thus it follows that  EMBED Equation.3  Given that r is a primitive root of p the order of r is p – 1. By the previous congruence  EMBED Equation.3 so p – 1 must divide m(p – 1)/2 which implies that m must be an even integer. Let m = 2k, then  EMBED Equation.3  making the integer  EMBED Equation.3 a solution of the congruence  EMBED Equation.3  Thus a is a quadratic residue modulo p which is what we wanted to show. ²% Example 4: Let a = 17. For which primes p is 17 a quadratic residue? We can test prime p's manually given the formula above. If we let p = 3, we have 17(3 " 1)/2 = 171 a" 2 (mod 3) a" -1 (mod 3), therefore 17 is not a quadratic residue modulo 3. Now if we let p = 13, we have 17(13 " 1)/2 = 176 a" 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 a" 4 (mod 13), and 22 = 4. ²% The following results are direct consequences of Euler s Criterion. Lemma 3: The congruence  EMBED Equation.3 is solvable if and only if  EMBED Equation.3  Proof: By Euler’s Criterion,  EMBED Equation.3 is solvable if and only if  EMBED Equation.3 . But  EMBED Equation.3  is either 1 or -1 depending on whether (p – 1)/2 is even or odd, respectively. Since p > 2,  EMBED Equation.3 if and only if (p – 1)/2 is even. But (p – 1)/2 is even if and only if  EMBED Equation.3  which is what we wanted to show. ²% Example 5: When p = 17, the congruence  EMBED Equation.3 is solvable since  EMBED Equation.3 . The solutions to the congruence  EMBED Equation.3 are exactly  EMBED Equation.3 . ²% Corollary 1: Let  EMBED Equation.3  be an odd prime and a any integer such that EMBED Equation.3 . Then a is a quadratic residue modulo p if and only – a is a quadratic nonresidue modulo p. Proof:  EMBED Equation.3  Let a be a quadratic residue modulo p. By Euler’s Criterion, if a is a quadratic residue modulo p then  EMBED Equation.3  So to show that –a is a quadratic nonresidue we need only show that  EMBED Equation.3  But  EMBED Equation.3  since (p – 1)/2 is odd. Hence – a is a quadratic nonresidue modulo p, which is what we wanted to show.  EMBED Equation.3  Let –a be a quadratic residue modulo p. By Euler’s Criterion, if –a is a quadratic residue modulo p then  EMBED Equation.3  So to show that a is a quadratic nonresidue we need only show that  EMBED Equation.3  But  EMBED Equation.3  since (p – 1)/2 is odd. Hence a is a quadratic nonresidue modulo p, which is what we wanted to show. ²% Corollary 2: Let  EMBED Equation.3  be an odd prime and a any integer such that EMBED Equation.3 . Then a and  a are either both quadratic residues modulo p or quadratic nonresidues modulo p. Proof: First we will assume that a be a quadratic residues modulo p. By Euler’s Criterion, if a is a quadratic residue modulo p then  EMBED Equation.3  So to show that –a is a quadratic residue modulo p we need only show that  EMBED Equation.3  But  EMBED Equation.3  since (p – 1)/2 is even. Hence –a is a quadratic residue modulo p. Now assume that a is a quadratic nonresidue modulo p. Then by Euler’s Criterion,  EMBED Equation.3  To Show that –a is also a quadratic nonresidue modulo p we need only show that  EMBED Equation.3  But  EMBED Equation.3  since (p  1)/2 is even. Hence  a is a quadratic nonresidue modulo p, which is what we wanted to show. ²% 3. The Law of Quadratic Reciprocity We are now ready to present the statement of the Law of Quadratic Reciprocity. Law of Quadratic Reciprocity: Let p and q be distinct odd primes. Then  EMBED Equation.3 . We should remark that by the statement of the Law of Quadratic Reciprocity  EMBED Equation.3  if  EMBED Equation.3 or EMBED Equation.3 , while  EMBED Equation.3  if  EMBED Equation.3 . Before we look at how we can actually prove this statement we will first look at a few examples which illustrate the different ways in which the Law of Quadratic Reciprocity may be used. Example 6: Let p = 3 and q = 17 then  EMBED Equation.3  which is what is expected since q = 17 is of the form 4k + 1. Thus by the Law of Quadratic Reciprocity the congruences  EMBED Equation.3  and  EMBED Equation.3  are either both solvable or both insolvable. The easiest congruence to check would be  EMBED Equation.3  since it reduces to the congruence  EMBED Equation.3  which is clearly not solvable since  EMBED Equation.3  and  EMBED Equation.3  are not divisible by 3. Therefore both of the congruences are insolvable and so  EMBED Equation.3  and EMBED Equation.3 . ²% The following examples illustrate how you can use the Law of Quadratic Reciprocity to determine the quadratic character of p with respect to q, when the value of either p or q is known. Example 7: In this example we will determine the value of the quadratic character of 7 with respect to p when p is an odd prime different from 7. We should note that 7 is an odd prime of the form 4k + 3 (since 7 = 4(1) + 3). Now by the Law of Quadratic Reciprocity we have  EMBED Equation.3  To determine the value of  EMBED Equation.3 we must consider two things: 1) The value of  EMBED Equation.3 ; 2) The value of the exponent on -1. To find the value of  EMBED Equation.3  we can find the least residues modulo 7 squaring the numbers from 1 to 3 EMBED Equation.3  and reducing them mod 7. Squaring the numbers from 1 to 3 yields  EMBED Equation.3 . Thus, the quadratic residues mod 7 are 1, 2, 4 and the quadratic nonresidues are 3, 5, 6. Now with the previous work done we can determine the value of  EMBED Equation.3  for each value of a from 1 to 6. If EMBED Equation.3 , then EMBED Equation.3  since 1 is a quadratic residue mod 7. If EMBED Equation.3 , then EMBED Equation.3  since 2 is a quadratic residue mod 7. If EMBED Equation.3 , then EMBED Equation.3  since 3 is a quadratic nonresidue mod 7. If EMBED Equation.3 , then EMBED Equation.3  since 4 is a quadratic residue mod 7. If EMBED Equation.3 , then EMBED Equation.3  since 5 is a quadratic nonresidue mod 7. If EMBED Equation.3 , then EMBED Equation.3  since 6 is a quadratic nonresidue mod 7. Next we must note that since p is an odd prime then p is either of the form 4k + 1 or 4k + 3. Case 1) p = 4k + 1 where k is any integer  EMBED Equation.3  If  EMBED Equation.3 , then the exponent on -1 is  EMBED Equation.3  an even number. Case 2) p = 4k + 3 where k is any integer  EMBED Equation.3  If  EMBED Equation.3 , then the exponent on -1 is  EMBED Equation.3  an odd number. The assumptions on p show that it is not divisible by 7 because we assumed that p was odd prime different from 7 and it is not divisible by 4 because it either has a remainder of 1 or 3 when divided by 4. Thus, p must be relatively prime to their product (7)(4) = 28. So p must be congruent to one of 1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, or 27 (all numbers between 1 and 28 that are relatively prime to 28). If  EMBED Equation.3 , then 28 | (p – 1)  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  the exponent on -1 is even so  EMBED Equation.3   EMBED Equation.3 . Hence the value of the Legendre symbol is  EMBED Equation.3  This example illustrates how when one of the odd primes is of the form 4k + 1, then either both of the congruences are solvable or both are unsolvable, yielding the same quadratic characters for  EMBED Equation.3 and EMBED Equation.3  In this case, they are both solvable since  EMBED Equation.3  Next we will illustrate the case when both p and q are of the form 4k + 3, if EMBED Equation.3 then  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3 the exponent -1 is odd so  EMBED Equation.3   EMBED Equation.3  (since 3 is a quadratic residue mod 7) Hence the value of the Legendre symbol for  EMBED Equation.3  In this example the value of the quadratic characters for  EMBED Equation.3 and EMBED Equation.3 are opposite, so one of the congruences is solvable and one is not. You could continue in this fashion checking all of the values 1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, and 27. The results are summarized below. 1 if  EMBED Equation.3   EMBED Equation.3 = -1 if  EMBED Equation.3  From this it is easy to see that if for example p = 27 then  EMBED Equation.3  Thus, there must exist an x such that  EMBED Equation.3 is solvable. After checking the values from 1 to 27 it can be easily be verified that  EMBED Equation.3 are the solutions to the congruence since  EMBED Equation.3  which is clearly true since 162 = (27)(6). ²% Example 8: What about for larger values of p? Suppose that p = 1,999 and q = 3. Both are of the form 4k + 3 so by The Law of Quadratic Reciprocity of the congruences  EMBED Equation.3 and  EMBED Equation.3 only one is solvable. The first congruence  EMBED Equation.3 reduces to  EMBED Equation.3  since 1,999 is congruent to 1 mod 3.  EMBED Equation.3  is clearly solvable when  EMBED Equation.3 since  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Therefore, by the Law of Quadratic Reciprocity the second congruence  EMBED Equation.3 must have no solution. You could verify that this congruence has no solution by testing each of the numbers from 1 to 999, but the Law of Quadratic Reciprocity solves the problem without any need for testing. Now let’s look at a case where one of the numbers p or q is of the form 4k + 1. Suppose that p = 1,999 and q = 5. Then the law states that the congruences  EMBED Equation.3  and  EMBED Equation.3  are either both solvable or unsolvable. The first congruence  EMBED Equation.3  reduces to  EMBED Equation.3  since 1,999 is congruent to 4 mod 5. The congruence  EMBED Equation.3 is clearly solvable when  EMBED Equation.3  since  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . By the Law of Quadratic Reciprocity the other congruence  EMBED Equation.3  must also be solvable. In order to determine the solution by brute force we would have to square the numbers between 1 and 999. It turns out that there is another method for finding the solution without squaring all of the numbers from 1 to  EMBED Equation.3  If p is of the form 4k + 3 (which in this case it is,  EMBED Equation.3  then one solution is  EMBED Equation.3  So for this particular example the congruence  EMBED Equation.3  has the solution  EMBED Equation.3  With the help of a calculator it can be checked that this yields  EMBED Equation.3 , so the other answer is  EMBED Equation.3 . ²% There are several other uses of Quadratic Reciprocity that may found in many textbooks. The reader may want to refer to Burton s text, Elementary Number Theory, for further illustrations. 4. Geometric Proof The geometric interpretation of Gauss s proof of the Law of Quadratic Reciprocity relies very heavily on what is known today as Gauss’s Lemma. So before we begin our exploration of this proof we will first introduce and discuss this concept. Lemma 4: Gauss’s Lemma: Let p be an odd prime and a an integer not divisible by p. Consider the least residues of the integers ka for  EMBED Equation.3 , reduced to lie between  EMBED Equation.3  to  EMBED Equation.3 . If the number of these which are negative is s, then EMBED Equation.3 . Proof: Let the least residues be  EMBED Equation.3  for  EMBED Equation.3  and  EMBED Equation.3  for  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3 , and  EMBED Equation.3  We first claim, with proof to follow, that the set EMBED Equation.3  Given that this is true we have  EMBED Equation.3  if we divide both sides by  EMBED Equation.3  we are left with  EMBED Equation.3 . Now by Euler’s Criterion EMBED Equation.3 . Thus we have that EMBED Equation.3 . Given that  EMBED Equation.3  and  EMBED Equation.3 will both yield values of either 1 or -1, it follows that  EMBED Equation.3  (since p is prime) which is what we wanted to show. Thus, all we need is to prove the claim that the set of least residues disregarding signs is exactly equal to the set of integers from 1 to  EMBED Equation.3 or equivalently  EMBED Equation.3  To do this we must show that no two  EMBED Equation.3  are congruent, no two  EMBED Equation.3 are congruent and that there are no  EMBED Equation.3  We will do this by contradiction. Let’s assume that two  EMBED Equation.3  are congruent then  EMBED Equation.3 with  EMBED Equation.3  and assuming without loss of generality that  EMBED Equation.3  (if  EMBED Equation.3  we would be talking about the same element in the set).  EMBED Equation.3   EMBED Equation.3  (since  EMBED Equation.3  This last statement is always false because both  EMBED Equation.3  and  EMBED Equation.3  fall within the range  EMBED Equation.3  So  EMBED Equation.3  Thus,  EMBED Equation.3  is too small to be divisible by p. Next we assume that two  EMBED Equation.3  are congruent then  EMBED Equation.3  with  EMBED Equation.3  Now  EMBED Equation.3   EMBED Equation.3  (since  EMBED Equation.3  This last statement is again false by the same reasoning as above. Finally, we will assume that  EMBED Equation.3  for some EMBED Equation.3  EMBED Equation.3 , then  EMBED Equation.3  for some  EMBED Equation.3  So  EMBED Equation.3   EMBED Equation.3  (since  EMBED Equation.3  This last statement is also false because again both  EMBED Equation.3  and  EMBED Equation.3  fall with the range  EMBED Equation.3  So  EMBED Equation.3  Clearly no integer within this range is divisible by p. Thus, we have shown that no two integers in the set  EMBED Equation.3  are congruent to one another and since there are exactly EMBED Equation.3  elements in this set, each within the range 1, . . . . ,  EMBED Equation.3 , it must be that EMBED Equation.3  Which is what we wanted to show. ²% Example 9: Take p = 11 and a = 19, we first reduce each 19k for  EMBED Equation.3 mod 11 to lie within the range  EMBED Equation.3  to EMBED Equation.3 . Thus, we get for k = 1,  EMBED Equation.3  for k = 2,  EMBED Equation.3  for k = 3,  EMBED Equation.3  for k = 4,  EMBED Equation.3  for k = 5,  EMBED Equation.3  So the least residues for the first 5 multiples of 19 modulo 11 are -1, 2, -3, -4, 5, which we should note are exactly the integers from 1 to EMBED Equation.3 . From the chart above we see there are 3 minus signs, so by Gauss’s Lemma EMBED Equation.3 . So 19 is a quadratic nonresidue mod11. Thus there does not exist an x such that EMBED Equation.3 . ²% In this next example we will just reverse the roles of 19 and 11. Example 10: Now let p = 19 and a = 11, we first reduce each 11k for  EMBED Equation.3 mod 11 to lie within the range  EMBED Equation.3  to  EMBED Equation.3 . Thus, we get for k = 1,  EMBED Equation.3  for k = 2,  EMBED Equation.3  for k = 3,  EMBED Equation.3  for k = 4,  EMBED Equation.3  for k = 5,  EMBED Equation.3  for k = 6,  EMBED Equation.3  for k = 7,  EMBED Equation.3  for k = 8,  EMBED Equation.3  for k = 9,  EMBED Equation.3  So the least residues for the first 9 multiples of 11 modulo 19 are 1, -2, 3, 4, -5, 6, -7, -8, 9, which in this case are exactly the set of integers from 1 to  EMBED Equation.3 . From the chart above we see that there are 4 minus signs, so by Gauss’s Lemma  EMBED Equation.3 . In this case, 11 is a quadratic residue mod 19. Thus, the congruence EMBED Equation.3 is solvable. In fact, the solutions to this congruence are exactly EMBED Equation.3  since  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  19 | 38 which is clearly true. A similar argument would hold for  EMBED Equation.3  ²% We are now ready to look at the geometric interpretation of Gauss s proof. The following proof is based on a proof presented in the The Jewel of Arithmetic: Quadratic Reciprocity found online at www. webpages.com. Proof of the Law of Quadratic Reciprocity: A point in the x, y plane will be called a lattice point if both its coordinates are integers. We will count lattice points in various regions. Consider the rectangle R in the x, y plane with sides y = 1 and  EMBED Equation.3  and vertical sides x = 1 and  EMBED Equation.3  The total number of lattice points in or on the rectangle is equal to  EMBED Equation.3  (exactly equal to the exponent on (-1) in the statement of the theorem). This is illustrated in the graph below where p = 11 and q = 19, in this case there are  EMBED Equation.3 lattice points in the rectangle. Figure 1  EMBED PBrush  Next we will use three lines to divide the rectangle into four regions. Let L be the line EMBED Equation.3  (or x =  EMBED Equation.3 )  EMBED Equation.3  be the line  EMBED Equation.3   EMBED Equation.3  be the line  EMBED Equation.3  (or  EMBED Equation.3 ) Figure 2  EMBED PBrush  We should note that there are no lattice points on any of these lines within the rectangle because  EMBED Equation.3  and  EMBED Equation.3  since all of the coordinates x within the rectangle are in the range  EMBED Equation.3  and all of the coordinates y are in the range  EMBED Equation.3  We will let A be the region above  EMBED Equation.3  where  EMBED Equation.3  B be the region below  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  be the region between L and  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  be the region between L and  EMBED Equation.3  where  EMBED Equation.3  This again is illustrated in the graph below for the case when p = 11 and q = 19. Figure 3  EMBED PBrush  If we let P(X) denote the number of lattice points in region X it should be that P(A) + P(B) + P( EMBED Equation.3 ) + P( EMBED Equation.3 ) =  EMBED Equation.3  Thus, the statement of the theorem could be written in the form  EMBED Equation.3  We will first show that A and B contain the same number of lattice points so the sum of P(A) and P(B) will always be even. Thus, they will have no effect on the exponent of (-1). So the statement of the theorem can be reduced to the following expression  EMBED Equation.3  We will then show that  EMBED Equation.3  contains  EMBED Equation.3  points where  EMBED Equation.3  and that  EMBED Equation.3  contains EMBED Equation.3  where EMBED Equation.3 . So we could again restate the theorem as  EMBED Equation.3  where, EMBED Equation.3 . To see that P(A) = P(B) note that the entire picture is symmetric about its center point  EMBED Equation.3  as illustrated in the graph below. Figure 4  EMBED PBrush  To verify this claim, let f :  EMBED Equation.3  be defined by  EMBED Equation.3  with the property that  EMBED Equation.3  We will show that this mapping f is a one-one correspondence between the lattice points of A and those in B. Proof: To prove that f is a one-one correspondence between the lattice points of A and B we must show that f is a one-one and onto mapping between the lattice points of A and B. We must first verify that f maps the lattice points of A to the lattice points in B. Let EMBED Equation.3  be any lattice point in A, then x and y are both integer values such that  EMBED Equation.3 . We wish to show that  EMBED Equation.3  is a lattice point in B. Observe first that f (x, y) will be a lattice point since both p and q were assumed to be odd. By the definitions of regions A and B, the reader can easily check that if  EMBED Equation.3  then  EMBED Equation.3  . Hence EMBED Equation.3 , which is what we wanted to show. To show that f is one-one we must show that f(a, b) = f (c, d) implies that (a, b) = (c, d). Now,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and b = d  EMBED Equation.3  which is what we wanted to show. Therefore, f is one-one. Next we must show that f is onto. To show that f is onto we must show for each  EMBED Equation.3  there exists an EMBED Equation.3 such that  EMBED Equation.3  Suppose  EMBED Equation.3  and let EMBED Equation.3 , then (x, y)  EMBED Equation.3 since by definition of the region B  EMBED Equation.3  and thus the reader can easily verify through algebra that this implies that  EMBED Equation.3 . Hence (x, y)  EMBED Equation.3  and (x, y) is clearly a lattice point since p and q were assumed to be odd. Further, f (x, y)  EMBED Equation.3  Thus, there exists a lattice point  EMBED Equation.3 , namely EMBED Equation.3 , such that  EMBED Equation.3  Therefore, f is onto. We have shown that f is one-one and onto. Therefore, we have shown that f is a one-one correspondence between the lattice points of A and those of B. Hence, P(A) = P(B). Next we will examine the number of lattice points in  EMBED Equation.3  Let  EMBED Equation.3  Then  EMBED Equation.3  if and only if  EMBED Equation.3  by definition of the region  EMBED Equation.3  The interval has length  EMBED Equation.3  so it contains at most one integer y. There will be an integer y in this interval if and only if there is an integer  EMBED Equation.3  (where  EMBED Equation.3 denotes the greatest integer) in the interval  EMBED Equation.3  which is if and only if  EMBED Equation.3  Multiplying through by p we get the equivalent statement  EMBED Equation.3 . The integer in the middle is the remainder on dividing qx by p. The number of x’s in the range  EMBED Equation.3  for which the remainder lies between  EMBED Equation.3  and p is exactly the same integer s which occurs in Gauss’s Lemma satisfying  EMBED Equation.3  A similar argument could be used to show that the number of lattice points in  EMBED Equation.3 is the integer  EMBED Equation.3  which appears in Gauss s Lemma such that  EMBED Equation.3  Thus we have,  EMBED Equation.3  which is exactly what we wanted to show. ²% Example 11 (for p = 11 and q = 19): We should note that p and q are both of the form 4k + 3. Thus, we must show that  EMBED Equation.3  We will proceed by counting lattice points in various regions. Consider the rectangular region R in the x, y plane with sides y = 1 and  EMBED Equation.3  and vertical sides x = 1 and  EMBED Equation.3 . The total number of lattice points in or on the rectangle is equal to EMBED Equation.3 , as illustrated in the graph below. Figure 5  EMBED PBrush  Next we will use three lines to divide the rectangle into four regions. We will let  EMBED Equation.3  be the line  EMBED Equation.3   EMBED Equation.3 be the line  EMBED Equation.3   EMBED Equation.3 be the line  EMBED Equation.3   EMBED Equation.3  We should note that no lattice points lie on any of these lines within the rectangle because  EMBED Equation.3  when  EMBED Equation.3 and similarly  EMBED Equation.3  when EMBED Equation.3 . We will let A be the region above  EMBED Equation.3  where  EMBED Equation.3  B be the region above  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3 be the region between L and  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3 be the region between L and  EMBED Equation.3  where  EMBED Equation.3  Figure 6  EMBED PBrush  If we let P(X) denote the number of lattice points in Region X, it should be that,  EMBED Equation.3  Thus, the statement of the theorem could be rewritten as  EMBED Equation.3  One could use a rigorous algebraic proof to show that the number P(A) = P(B). However, for the purpose of this discussion we will use the graph to illustrate this idea. Figure 7  EMBED PBrush  It is obvious from the graph that if you were to reflect region A about the line x = 3 and then again about the line y = 5, region A would lie right on top of region B. Thus, each lattice point in region A would correspond to exactly one lattice point in region B. Therefore, since f is one-one and onto it must be that P(A) = P(B). So the sum of their lattice points will always equal an even number. Thus, they will have no effect on the exponent -1 in the statement of the theorem. So we can omit them from the exponent leaving  EMBED Equation.3 . Next we will show that the region  EMBED Equation.3  contains  EMBED Equation.3  lattice points where  EMBED Equation.3  by showing that  EMBED Equation.3  is the exact same number that occurred in Gauss’s Lemma when p = 11 and q = 19. Similarly, we will show that  EMBED Equation.3  contains  EMBED Equation.3  lattice points where  EMBED Equation.3  by showing that  EMBED Equation.3  is the exact same number that occurred in Gauss’s Lemma when p = 19 and q = 11. So we would have  EMBED Equation.3  where  EMBED Equation.3  must be congruent to  EMBED Equation.3 mod 2. Let  EMBED Equation.3 , by definition  EMBED Equation.3  Since the length of the interval is only œ it can only contain at most one integer y. There will be an integer y in this interval if and only if there is an integer  EMBED Equation.3  in the interval  EMBED Equation.3 . which is if and only if  EMBED Equation.3  or multiplying through by 11 we get the equivalent statement  EMBED Equation.3  The integer in the middle,  EMBED Equation.3 , is the remainder on diving 19x by 11. Computing these remainders for the integers x in the range  EMBED Equation.3 , yields for x = 1,  EMBED Equation.3  for x = 2,  EMBED Equation.3  for x = 3,  EMBED Equation.3  for x = 4,  EMBED Equation.3  for x = 5,  EMBED Equation.3  Thus, there are only three integers that lie within the range  EMBED Equation.3  and 11, so  EMBED Equation.3  = 3. If we recall from example 3, that the least residues for 19 with respect to 11 were -1, 2, -3, -4, and 5, so in this case s = 3 (remember that s is the number of negative least residues). We have  EMBED Equation.3 = s = 3. Thus, the number of lattice points in EMBED Equation.3  is the exact same number which occurred in Gauss’s Lemma for  EMBED Equation.3  By counting the number of lattice points in region  EMBED Equation.3  on the graph we see there are indeed 3. So  EMBED Equation.3  By using a similar argument we will show that the number of lattice points in  EMBED Equation.3  is the same as the number of minus signs we found earlier using Gauss’s Lemma for p = 19 and q = 11. Let  EMBED Equation.3  by definition  EMBED Equation.3  if and only if  EMBED Equation.3  The interval again is only of length œ, so it will contain at most one integer x. There will be an integer x in this interval if and only if there is an integer  EMBED Equation.3  in the interval  EMBED Equation.3  which is true if and only if  EMBED Equation.3  Multiplying through by 19 we obtain the equivalent statement  EMBED Equation.3  The integer in the middle here is the remainder on dividing 11y by 19. Computing these remainders for the integers y in the range  EMBED Equation.3  we see for y = 1,  EMBED Equation.3  for y = 2,  EMBED Equation.3  for y = 3,  EMBED Equation.3  for y = 4,  EMBED Equation.3  for y = 5,  EMBED Equation.3  for y = 6,  EMBED Equation.3  for y = 7,  EMBED Equation.3  for y = 8,  EMBED Equation.3  for y = 9,  EMBED Equation.3  As we see there are only four integers that lie within the range  EMBED Equation.3  to 19, so  EMBED Equation.3  Again if we recall from example 4 that the least residues for 19 with respect to 11 were 1, -2, 3, 4, -5, 6, -7, -8, and 9 so in this case s = 4. So we have  EMBED Equation.3  Thus, the number of lattice points in region  EMBED Equation.3 corresponds exactly to the number which occurred in Gauss’s Lemma for  EMBED Equation.3  Counting the number of lattice points on the graph we see that region  EMBED Equation.3  does contain four lattice points. So  EMBED Equation.3  Therefore, we have  EMBED Equation.3  which is exactly what we wanted to show. ²% 5. Schmidt’s Proof We are now ready to begin our examination of Hermann Schmidt’s proof. As stated in the introduction of this paper Schmidt’s proof uses some of the more elementary results in Number Theory such as the Chinese Remainder Theorem and Wilson’s Theorem. So we begin our discussion with these and a few related results. Lemma 5: If (p, q) = 1, then  EMBED Equation.3  if and only if  EMBED Equation.3 and  EMBED Equation.3 . Proof:  EMBED Equation.3 Let p and q be two integers such that (p, q) = 1 and suppose that there exists integers x and a such that  EMBED Equation.3  By definition of congruence if  EMBED Equation.3 then EMBED Equation.3 . But if  EMBED Equation.3  then  EMBED Equation.3  and  EMBED Equation.3  and so  EMBED Equation.3  and  EMBED Equation.3 which is what we wanted to show.  EMBED Equation.3  Let p and q be two integers such that (p, q) = 1 and suppose that there exists integers x and a such that  EMBED Equation.3 and EMBED Equation.3 . If  EMBED Equation.3 then by definition of congruence  EMBED Equation.3 which implies that there exists and integer r such that (x – a) = rp. Similarly, if  EMBED Equation.3 then by definition of congruence  EMBED Equation.3 which implies that there exists an integer s such that (x – a) = sq. Now if (x – a) = rp and (x – a) = sq then rp = sq and so p must divide sq. But if  EMBED Equation.3 then  EMBED Equation.3 since it was assumed that (p, q) = 1. Hence there must exist an integer t such that tp = s. Thus (x  a) = sq = t(pq) which implies that  EMBED Equation.3  and so  EMBED Equation.3  which is what we wanted to show. ²% Recall the Chinese Remainder Theorem, which assures the existence of a solution to any system of linear congruences over moduli that are pairwise relatively prime. In this paper, we only require a special case involving a pair of congruences, as stated in the next theorem. We include a proof for the sake of completeness. Theorem 7: (Chinese Remainder Theorem): Let p and q be positive integers, with (p, q) = 1. Then the system of congruences  EMBED Equation.3   EMBED Equation.3  has a solution. Furthermore, any two solutions will be congruent modulo pq. Proof: Let p and q be two positive integers such that (p, q) = 1. Since (p, q) = 1, by the Euclidean Algorithm there must exist integers r and s such that rp + sq = 1. It follows that the integers sq and rp satisfy  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and so x = asq + brp must be a solution to the system of congruences EMBED Equation.3 and  EMBED Equation.3 . This can be seen by reducing modulo p and then modulo q as follows,  EMBED Equation.3  and  EMBED Equation.3  Thus, if (p, q) = 1 the system of congruences  EMBED Equation.3 and  EMBED Equation.3 will have a solution. If x is a solution, then adding any multiple of pq to x will still be a solution since  EMBED Equation.3 for any integer k. Conversely if  EMBED Equation.3  and  EMBED Equation.3 are two solutions of the given congruences, then they must be congruent modulo p and modulo q, that is if  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  then  EMBED Equation.3  and EMBED Equation.3 . Thus  EMBED Equation.3  is divisible by both p and q and so it is divisible by pq since by assumption (p, q) = 1. Therefore  EMBED Equation.3 which is what we wanted to show. ²% Example 12: To solve the system  EMBED Equation.3   EMBED Equation.3  we first use the Euclidean Algorithm to write (2)(3) + (-1)(5) = 1. Then x = (2)(3)(4) + (-1)(5)(1) = 19 is a solution and the general solution is x = 19 + 15t. The smallest nonnegative solution is therefore 4, so we have EMBED Equation.3 . ²% Lemma 6: Let p and q be positive integers, with (p, q) = 1. If a is a quadratic residue modulo p and a is a quadratic residue modulo q then a is a quadratic residue modulo pq. Proof: Suppose that a is a quadratic residue modulo p and modulo q. If a is a quadratic residue modulo p then there exists an x such that  EMBED Equation.3 . Considering the list of integers x, x + p, x + 2p, . . . . . , x + (q – 1)p we have that  EMBED Equation.3  in some order. Since a is also a quadratic residue modulo q at least one of the numbers from this second list, call it y, when squared, must yield a modulo q. That is,  EMBED Equation.3  But y must be congruent to one of the numbers in the list x, x + p, x + 2p, . . . . . , x + (q – 1)p. In other words,  EMBED Equation.3  for some integer i. Now we have that  EMBED Equation.3  and also  EMBED Equation.3  If  EMBED Equation.3 then  EMBED Equation.3  Thus we have that  EMBED Equation.3  and  EMBED Equation.3 so EMBED Equation.3  by Lemma 5. Hence a is quadratic residue modulo pq. ²% Next we would like to recall Wilson s Theorem which allows us to determine for what integers p, p is a prime. Theorem 8: (Wilson s Theorem): Let p be an integer. Then p is a prime if and only if  EMBED Equation.3  Proof: If p = 2, then  EMBED Equation.3 as desired. So assume that p is an odd prime. For the rest of the proof we will assume that p is an odd prime.  EMBED Equation.3 If p is an odd prime and G = {1, 2, . . . . , p – 1} then by Lemma 2 each a in G will have a unique inverse modulo p. If EMBED Equation.3 , then multiplying both sides of this congruence by a yields  EMBED Equation.3 . If  EMBED Equation.3 then  EMBED Equation.3  or equivalently EMBED Equation.3 . Given that p is prime,  EMBED Equation.3 implies  EMBED Equation.3 and so a = 1 or a = p – 1. Thus, 1 and p – 1 are each their own inverses, but every other element of G has a distinct inverse, and so if we collect the elements of G pairwise and multiply them all together we have  EMBED Equation.3  but  EMBED Equation.3  Hence  EMBED Equation.3 .  EMBED Equation.3  Now suppose by contradiction that the congruence holds for a composite number p. If p is composite then p has a proper divisor d with 1 < d < p. If d < p then d also divides  EMBED Equation.3  since d must be one of the integers between 1 and p – 1. However, we assumed the congruence  EMBED Equation.3 holds so d must also divide (p – 1)! + 1 and so d must divide 1 which a contradiction. Thus if the congruence holds p must be prime which is what we wanted to show. ²% Example 13: Let p = 11, then we should be able to pair the integers from 1 to 10 EMBED Equation.3 , not inclusive, with their inverses. Indeed,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  Thus,  EMBED Equation.3  ²% With Wilson s Theorem and the Chinese Remainder Theorem established we will begin our exploration of Schmidt s proof, which may be found in his paper entitled Drei neue Beweise des Reciprocitätssates in der Theorie der quadratischen Reste, beginning with the following lemma. Lemma 7: Let p, q be distinct odd primes. If r denotes any quadratic residue (mod pq), then the congruence  EMBED Equation.3 has exactly four distinct solutions,  EMBED Equation.3  through  EMBED Equation.3  that satisfy: (1)  EMBED Equation.3  (2)  EMBED Equation.3  In particular, two solutions have positive least residues modulo pq. Proof: Assume r is a quadratic residue mod pq. Then the congruence  EMBED Equation.3 has solutions mod pq. So by Lemma 5, this congruence has solutions mod p and mod q. Indeed, by Proposition 1,  EMBED Equation.3 has exactly two incongruent solutions mod p, call them c and –c. Similarly,  EMBED Equation.3  has exactly two incongruent solutions mod q, call them d and – d. Therefore  EMBED Equation.3 must have exactly four solutions  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 , each satisfying a pair of congruences as follows: (i)  EMBED Equation.3  and  EMBED Equation.3  (ii)  EMBED Equation.3  and  EMBED Equation.3  (iii)  EMBED Equation.3  and  EMBED Equation.3  (iv)  EMBED Equation.3  and  EMBED Equation.3  By the Chinese Remainder Theorem  EMBED Equation.3  through  EMBED Equation.3  are each uniquely determined (mod pq). Moreover,  EMBED Equation.3  through  EMBED Equation.3  are all distinct (mod pq) by construction. Observe that  EMBED Equation.3 and  EMBED Equation.3  modulo pq and so the conditions of the Lemma are satisfied. ²% Definition 6: If r is any quadratic residue modulo pq then we say the four solutions of  EMBED Equation.3  are conjugates. Corollary 3: When the p and q are distinct odd primes then the number of quadratic residues modulo pq is exactly EMBED Equation.3 . Proof: By the above lemma, each of the (p – 1)(q – 1) residues prime to pq belongs to a set of four conjugates whose squares are congruent modulo pq to a common quadratic residue. It follows that there are exactly  EMBED Equation.3  quadratic residues modulo pq. ²% Remark By the Lemma 7, it follows that for odd primes EMBED Equation.3 , there will be four solutions to  EMBED Equation.3 . If we consider the least residues of these solutions, two will be positive and two will be negative. Clearly EMBED Equation.3  is one of the positive solutions. We henceforth will use  EMBED Equation.3  to denote the remaining positive solution. Lemma 8: Let p, q denote distinct odd primes. Let  EMBED Equation.3  through  EMBED Equation.3  represent the four conjugates stated in Lemma 7 satisfying: (1)  EMBED Equation.3  (2)  EMBED Equation.3  Let a and b be two remainders such that  EMBED Equation.3  and  EMBED Equation.3 . Let r be any quadratic residue modulo pq. If EMBED Equation.3 , then exactly one of the four solutions of  EMBED Equation.3  is itself a quadratic residue, the other three are not. If  EMBED Equation.3  and  EMBED Equation.3 (or visa versa) then either exactly two of the four solutions of  EMBED Equation.3  are quadratic residues or else none of them are. If  EMBED Equation.3 then either all four solutions of  EMBED Equation.3 are quadratic residues, or else none of them are. Proof (i): Let p and q be odd primes such that EMBED Equation.3 . Then by Corollary 1, either a or –a is a quadratic residue modulo p. Similarly either b or –b is a quadratic residue modulo q. Thus we have the following four cases: Case 1: (a and b are both quadratic residues) Assume that a and b are quadratic residues modulo p and q respectively. If a is a quadratic residue modulo p then by definition of a and (1) above,  EMBED Equation.3  and  EMBED Equation.3 are both quadratic residues modulo p. Similarly if b is a quadratic residue modulo q then by definition of b and (2) above,  EMBED Equation.3  and  EMBED Equation.3  are quadratic residues modulo q. Hence  EMBED Equation.3  is a quadratic residue modulo pq by Lemma 6 and the others are not. Case 2: (- a and b are quadratic residues) Assume that –a and b are quadratic residues modulo p and q respectively. If –a is a quadratic residue modulo p then  EMBED Equation.3  and  EMBED Equation.3  are both quadratic residues modulo p by definition (1) above. If b is a quadratic residue modulo q then EMBED Equation.3 and  EMBED Equation.3  are both quadratic residues modulo q by definition (2) above. Hence EMBED Equation.3  is a quadratic residue modulo pq by Lemma 6 and the others are not. Case 3: (a and –b are quadratic residues) Assume that a and –b are quadratic residues modulo p and q respectively. If a is a quadratic residue modulo p then  EMBED Equation.3  and  EMBED Equation.3  are both quadratic residues modulo p by definition (1) above. If –b is a quadratic residue modulo q then EMBED Equation.3 and  EMBED Equation.3  are both quadratic residues modulo q by definition (2) above. Hence EMBED Equation.3  is a quadratic residue modulo pq by Lemma 6 and the others are not. Case 4: (– a and –b are quadratic residues) Assume that –a and –b are quadratic residues modulo p and q respectively. If –a is a quadratic residue modulo p then  EMBED Equation.3  and  EMBED Equation.3  are both quadratic residues modulo p by definition (1) above. If –b is a quadratic residue modulo q then EMBED Equation.3 and  EMBED Equation.3  are both quadratic residues modulo q by definition (2) above. Hence EMBED Equation.3  is a quadratic residue modulo pq by Lemma 6 and the others are not. Thus we have shown that when  EMBED Equation.3  in each of these cases only one of the solutions  EMBED Equation.3 through  EMBED Equation.3  is a quadratic residue modulo pq, which is what we wanted to show. Proof (ii): Without loss of generality, assume EMBED Equation.3 and  EMBED Equation.3 . If  EMBED Equation.3 then a and –a are either both quadratic residues or quadratic nonresidues modulo p. Thus we have the following two cases: Case 1: (a, –a and b or –b are quadratic residues) Assume that a and –a are both quadratic residues modulo p then by definition (1) we have that EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 are all quadratic residues modulo p. Now since EMBED Equation.3 , if b is a quadratic residue modulo then by Corollary 1, – b is a quadratic nonresidue and visa versa. If b is a quadratic residue modulo q then by definition (2) above  EMBED Equation.3  and  EMBED Equation.3  are quadratic residues modulo q. Thus by Lemma 6,  EMBED Equation.3 and  EMBED Equation.3  would be quadratic residues modulo pq and  EMBED Equation.3 and  EMBED Equation.3 would not. Similarly if – b is a quadratic residue modulo q then by definition (2) above  EMBED Equation.3  and  EMBED Equation.3 are quadratic residues modulo pq. Thus by Lemma 6,  EMBED Equation.3 and  EMBED Equation.3 would be quadratic residues modulo pq and  EMBED Equation.3 and  EMBED Equation.3 would not. In either case exactly two of the solutions  EMBED Equation.3  through EMBED Equation.3  are quadratic residues modulo pq. Case 2: (a and –a are quadratic nonresidues) Assume that a and –a are quadratic nonresidues modulo p then  EMBED Equation.3  through EMBED Equation.3  are quadratic nonresidues modulo p. Therefore EMBED Equation.3  through EMBED Equation.3  are quadratic nonresidues modulo pq. Thus we have shown that if  EMBED Equation.3  and  EMBED Equation.3 (or visa versa) then exactly two of the conjugates are quadratic residues modulo pq or none of them are. Which is what we wanted to show. Proof (iii): Let EMBED Equation.3 . If  EMBED Equation.3 then both a and – a are either both quadratic residues or both quadratic nonresidues modulo p. Similarly, if  EMBED Equation.3 both b and – b are either both quadratic residues or both quadratic nonresidues modulo q. Thus we have the following two cases: Case 1: (a, – a , b, – b are all quadratic residues) Assume that a, –a are quadratic residues modulo p and b, – b are quadratic residues modulo q. Then all four solutions are quadratic residues modulo p and modulo q. Hence by Lemma 6, all four solutions would be quadratic residues modulo pq. Case 2: (a, – a quadratic nonresidues or b, – b quadratic nonresidues) Without loss of generality assume that a and –a are quadratic nonresidues modulo p. Then all of the four solutions  EMBED Equation.3  through EMBED Equation.3 are quadratic nonresidues modulo p. Hence all four solutions are quadratic residues modulo pq. We have shown that if  EMBED Equation.3 then all four of the solutions are quadratic residues modulo pq or none of them are, which is what we wanted to show. ²% Example 14: This example illustrates the case when both p and q are congruent to 3 modulo 4. Let p = 3 and q = 7 then the (p  1)(q  1) =(2)(6) = 12 remainders prime to 21 are 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20. Squaring each of these numbers and reducing them modulo 21 we have  EMBED Equation.3 {– 1, 1, – 8, 8} solve  EMBED Equation.3   EMBED Equation.3 {– 2, 2, – 5, 5} solve  EMBED Equation.3   EMBED Equation.3 {– 4, 4, – 10, 10} solve  EMBED Equation.3  So we can easily see that there are exactly four remainders, two negative and two positive, whose squares are congruent modulo 21. Observe that each set of four contains exactly one quadratic residue. If we choose the remainders – 5 and 5 then –a = - 2 , a = 2, –b = - 5 and b =5. Thus our four conjugates will be  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  Since p = 3 is congruent to 3 mod 4 and so only one of the numbers – 2 or 2 is a quadratic residue modulo 3. If we square the numbers 1 and 2 we see  EMBED Equation.3  and  EMBED Equation.3  and so – 2 is a quadratic residue modulo 3 and 2 is not. Similarly q = 7 is congruent to 3 mod 4 and so only one of the numbers – 5 or 5 is a quadratic residue modulo 7. If we square the numbers from 1 to 6 and reduce each modulo 7 we have  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  and so – 5 is a quadratic residue modulo 7 and 5 is not. Thus the conjugate  EMBED Equation.3  is the only quadratic residue modulo 21. If  EMBED Equation.3  and  EMBED Equation.3 then  EMBED Equation.3  and – 5 is a quadratic residue modulo 21 since  EMBED Equation.3 and  EMBED Equation.3  ²% Example 15: This example illustrates the case when either p or q is congruent to 1 modulo 4. Let p = 3 and q = 5 then the (p  1)(q  1) =(2)(4) = 8 remainders prime to 15 are 1, 2, 4, 7, 8, 11, 13, 14. Squaring each of these numbers and reducing them modulo 15 we have  EMBED Equation.3 {– 1, 1, – 4, 4} solve  EMBED Equation.3   EMBED Equation.3 {– 2, 2, – 7, 7} solve  EMBED Equation.3  So we can easily see that there are exactly four remainders, two negative and two positive, whose squares are the same or congruent modulo 15. Observe that the first set of four contains exactly two quadratic residues and the other does not contain any. If we choose the remainders – 4 and 4 then –a = - 1, a = 1, –b = - 4 and b =4. Thus our four conjugates will be  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  Since p = 3 is congruent to 3 mod 4 only one of the numbers – 1 or 1 is a quadratic residue modulo 3. If we square the numbers 1 and 2 we see  EMBED Equation.3  and  EMBED Equation.3  and so 1 is a quadratic residue modulo 3 and -1 is not. Now q = 5 is congruent to 1 mod 4 and so either the numbers – 4 and 4 are both quadratic residues or both quadratic nonresidues modulo 5. If we square the numbers from 1 to 4 and reduce each modulo 5 we have  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  and so – 4 and 4 are both quadratic residues modulo 5. Thus the conjugates  EMBED Equation.3 and  EMBED Equation.3 must be quadratic residues modulo 15. If  EMBED Equation.3  and  EMBED Equation.3 then  EMBED Equation.3  and 4 is a quadratic residue modulo 15 since EMBED Equation.3  If  EMBED Equation.3  and  EMBED Equation.3 then we can determine the quadratic residue modulo 15 by using the method implied by the proof of the Chinese Remainder Theorem. By the Euclidean algorithm (3)(2) + (5)(-1) =1 and so a solution to the system is x = (3)(2)(-4) + (5)(-1)(1) = -29 and so the smallest nonnegative solution EMBED Equation.3 . Thus 1 is a quadratic residue modulo 15 since  EMBED Equation.3  ²% Example 16: This example illustrates the case when both p and q are congruent to 1 modulo 4. Let p = 5 and q = 13 then squaring the (p  1)(q  1) =(4)(12) = 48 remainders prime to 65 and reducing each of modulo 65 we have {– 1, 1, – 14, 14} solve  EMBED Equation.3  {– 8, 8, – 18, 18} solve  EMBED Equation.3  {– 2, 2, – 28, 28} solve  EMBED Equation.3  {– 11, 11, – 24, 24} solve  EMBED Equation.3  {– 3, 3, – 23, 23} solve  EMBED Equation.3  {– 12, 12, – 27, 27} solve  EMBED Equation.3  {– 4, 4, – 9, 9} solve  EMBED Equation.3  {– 16, 16, – 29, 29} solve  EMBED Equation.3  {– 6, 6, – 19, 19} solve  EMBED Equation.3  {– 21, 21, – 31, 31} solve  EMBED Equation.3  {– 7, 7, – 32, 32} solve  EMBED Equation.3  {– 17, 17, – 22, 22} solve  EMBED Equation.3  So we can easily see that there are exactly four remainders, two negative and two positive, whose squares are the same or congruent modulo 65. Observe also that each set of four contains either exactly four or zero quadratic residues. If we choose the remainders – 14 and 14 then –a = - 4 , a = 4, –b = - 1 and b =1. Thus our four conjugates will be  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  Since p = 5 is congruent to 1 mod 4 either both of the numbers –4 and 4 are quadratic residues modulo 5 or both quadratic nonresidues modulo 5. 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-6]jâ?h¥}\hê -EHöÿUjUe~F hê -CJUVaJjˆ=h¥}\hê -EHôÿUj6e~F hê -CJUVaJjV;h¥}\hê -EHöÿUj%e~F hê -CJUVaJhê -jù8h¥}\hê -EHöÿUje~F hê -CJUVaJ h¥}\hê -jh¥}\hê -UjÂ6h¥}\hê -EHöÿU#ÞGßGàGáGåGæGùGúGûGüGýGþGH@@@@@@@)@*@+@,@-@o@p@OAPAcAdAðãØÔØÍŸ±ØÔØͯ͠“ØÔØ̈́wØpÔiÔØÍZjÀf~F hê -CJUVaJ hê -6] hSlhê -jIh¥}\hê -EHöÿUj>f~F hê -CJUVaJjŸFh¥}\hê -EHöÿUjf~F hê -CJUVaJUjuDh¥}\hê -EHöÿUj.f~F hê -CJUVaJ h¥}\hê -hê -jh¥}\hê -UjBh¥}\hê -EHöÿUjêe~F hê -CJUVaJMBED Equation.3   EMBED Equation.3  and so – 4 and 4 are both quadratic residues modulo 5. Similarly q = 13 is congruent to 1 mod 4 and so either both of the numbers – 1 and 1 are quadratic residues modulo 13 or both are quadratic nonresidues modulo 13. If we square the numbers from 1 to 6 and reduce each modulo 7 we have  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  and so –1 and 1 are both quadratic residues modulo 13. Thus all four conjugates must be quadratic residues modulo 65. If  EMBED Equation.3  and  EMBED Equation.3 then by the Euclidean Algorithm 1 = (2)(13) + (-5)(5) and so a solution to the system is x = (2)(13)(4) + (-5)(5)(1) = 79 which implies that the smallest nonnegative solution is  EMBED Equation.3 . Thus 14 is a quadratic residue modulo 65. If  EMBED Equation.3  and  EMBED Equation.3 then by the C.R.T a solution to this system is x = (2)(13)(4) + (-5)(5)(-1) = 129 which implies that the smallest nonnegative solution is  EMBED Equation.3 . Thus 64 is a quadratic residue modulo 65. If  EMBED Equation.3  and  EMBED Equation.3 then by the C.R.T. a solution to this system is x = (2)(13)(-4) + (-5)(5)(1) = -129 which implies that the smallest nonnegative solution is  EMBED Equation.3  and 1 is a quadratic residue modulo 65. Finally, if  EMBED Equation.3  and  EMBED Equation.3  then by the C.R.T. a solution to this system is x = (2)(13)(-4) + (-5)(5)(-1) = -79 which implies that the smallest negative solution is  EMBED Equation.3  and 51 is also a quadratic residue modulo 65. ²% Throughout the discussion of the next few lemmas and corollary we will be examining and referring to a particular product which we will define here. Definition 7: We will define EMBED Equation.3  to be the product of positive remainders that are prime to pq. That is,  EMBED Equation.3  Lemma 9: Let p and q be distinct odd primes. If EMBED Equation.3  then EMBED Equation.3  EMBED Equation.3 . Otherwise,  EMBED Equation.3  and  EMBED Equation.3  EMBED Equation.3 . Proof (i): If we assume that both p and  EMBED Equation.3 then both +1 and -1 are quadratic residues modulo p and modulo q. So by Lemma 6, – 1 is a quadratic residue modulo pq. Thus by Lemma 7 we have four remainders modulo pq whose squares are congruent – 1 (mod pq), of these, two are positive, we will denote one by EMBED Equation.3 , and the other by  EMBED Equation.3 , where  EMBED Equation.3  Thus we have four remainders 1,  EMBED Equation.3  EMBED Equation.3  and  EMBED Equation.3 whose squares and product are  EMBED Equation.3  For each of the remaining positive remainders a of pq there exists a unique remainder b such that  EMBED Equation.3  and so the entire product  EMBED Equation.3  Proof (ii): In the case where (p – 1)/2 or (q – 1)/2 is odd, that is when either p or q is congruent to 3 modulo 4, then -1 is a quadratic nonresidue modulo pq. Grouping all of the remaining positive remainders we have  EMBED Equation.3 ,  EMBED Equation.3 , etc. when reduced modulo pq. Thus, in all of the remaining cases we have that  EMBED Equation.3  ²% Corollary 4: Let p and q be distinct odd primes. Then  EMBED Equation.3  Proof: The exponent  EMBED Equation.3  is even if and only if p and  EMBED Equation.3 . So this corollary holds by the work above. ²% Lemma 10: Let p and q be distinct odd primes. Then  EMBED Equation.3  Proof: In order to determine the signs on  EMBED Equation.3  and  EMBED Equation.3  we will proceed by looking at the following product:  EMBED Equation.3  (1) If we consider this product modulo p then we obtain  EMBED Equation.3  Now by Wilson’s Theorem we have that  EMBED Equation.3 and so  EMBED Equation.3  Hence  EMBED Equation.3  Comparing this product to  EMBED Equation.3  we note that both products contain the same number of factors, namely (p – 1)  EMBED Equation.3 . We can derive the product in (1) from the product  EMBED Equation.3  by multiplying by  EMBED Equation.3  and dividing by  EMBED Equation.3 . That is,  EMBED Equation.3   EMBED Equation.3  But  EMBED Equation.3  Thus  EMBED Equation.3  or equivalently  EMBED Equation.3  or  EMBED Equation.3  or multiplying both sides by  EMBED Equation.3  we have (I)  EMBED Equation.3 . If we were to do the same analysis interchanging the roles of p and q we would obtain the congruence (II)  EMBED Equation.3 . If we multiply (I) and (II) we obtain  EMBED Equation.3  Which is what we wanted to show. ²% The result of the Law of Quadratic Reciprocity thus follows directly, as follows. Proof of the Law of Quadratic Reciprocity: By Corollary 3 we have that  EMBED Equation.3  and by the previous Lemma 8 we have just shown that  EMBED Equation.3  Thus we have that  EMBED Equation.3  EMBED Equation.3 . Dividing both sides of this equality by  EMBED Equation.3  and subtracting exponents, the reader could easily verify that this expression simplifies to  EMBED Equation.3 . Which is what we wanted to show. ²% Example 17: This next example illustrates Lemma 7 for the case where p = 5 and q =13. Let p = 5 and q = 13. Then  EMBED Equation.3   EMBED Equation.3  Given that 5 and 13 are both congruent to 1 modulo 4 we know that both -1 and 1 are quadratic residues modulo 65. We can see this is true by the congruences below  EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  Thus we have the four positive remainders 1,  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 = 1, 14, 8,  EMBED Equation.3  whose product  EMBED Equation.3  Now we will pair each of the following positive remainders with their unique inverses as follows:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Thus considering the product  EMBED Equation.3  modulo 65 yields  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  (since  EMBED Equation.3 )  EMBED Equation.3 . ²% Example 18: This next example illustrates Lemma 7 for the case when either p or q is congruent to 3 modulo 4. Let p = 5 and q = 11. Then  EMBED Equation.3   EMBED Equation.3  Given that 11 is congruent to 3 modulo 4 we know that 1 is a quadratic residue modulo 55 and -1 is not. We can easily see from the following congruences that 1 is a quadratic residue modulo 55  EMBED Equation.3  and  EMBED Equation.3 . Thus we have the two positive remainders 1 and  EMBED Equation.3 = 1 and 21 whose squares are  EMBED Equation.3  Now we will pair each of the following positive remainders with their unique inverses as follows:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Thus considering the product  EMBED Equation.3  modulo 55 yields  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . Now  EMBED Equation.3  and  EMBED Equation.3 . Thus  EMBED Equation.3  and so exactly one of the congruences  EMBED Equation.3 and  EMBED Equation.3 and the other is not. It might be easiest to check the congruence EMBED Equation.3 since it reduces to  EMBED Equation.3 . By Lemma 3, the congruence  EMBED Equation.3  is solvable since  EMBED Equation.3 . Therefore, by the Law of Quadratic Reciprocity, the congruence  EMBED Equation.3  is not solvable. ²% 6. Conclusion The Law of Quadratic Reciprocity has been an important theme in the development of number theory. Since the time of Gauss, many other mathematicians such as Cauchy, Jacobi, and Kronecker have all devised their own proofs of the law to determine for themselves why such a deep reciprocal relationship should exist between simple congruences linking any pair of odd primes. To date there are over 200 published proofs for the Law of Quadratic Reciprocity. As was mentioned earlier Gauss provided several proofs himself, eight to be exact, whicdAeAfAkAlAA€AA‚AƒA„A—A˜A™AšAŸA A³AŽAµA¶A·AžAËAÌAÍAÎAÏAÐAãAäAåAòçãçÜÍÀçãçܱ€çãçܕˆçãçÜylçãçÜ]Pj Wh¥}\hê -EHöÿUjXg~F hê -CJUVaJjŽTh¥}\hê -EHöÿUj*\]hÙ56>*\]hÉ3Ï56>*\]hÙhØ_Œjêrh¥}\hê -EHöÿUjn~F hê -CJUVaJ hê -6]jph¥}\hê -EHôÿUj.n~F hê -CJUVaJ h¥}\hê -jh¥}\hê -Uhê -ÀHÁHÔHÕHÖH×HØHÞHßHàHâHæHçHìHíHI I III I!I&I'I:I;IIQI÷óä×÷Ðź¯š€€€–Ž€rŽ€Ž€cVŽ€j~€h†=*hEHòÿUjóœF hCJUVaJj2~hhEHöÿUjOòœF hCJUVaJjhU hh h6]h hØ_Œ5\hØ_Œ56>*\]h·II56>*\]hê -56>*\] hÙhÙj0whr•hÉ3ÏEHfÿUjËøF hÚu¿CJUVaJhÙjhÙUBHØHIVIaI­I®I®L¯LRNVNXNÈNþN"P6P¢PØPR÷ïääØØÌÌÌÄÄ¿·ÌIJ·Ìgd·IIdàgd‚#ÞgdHdàgdØ_Œ „hdh^„hgdØ_Œ „8dh^„8gdØ_Œ & FdhgdØ_Œdhgd·IIdhgdØ_ŒQIRISITIaIbIuIvIwIxI}I~I‘I’I“I•IšI©IªI«I­I®I³I·IºIÑIÒI×IØIëIìIðãÛ×Û×È»Û×Û׬ŸÛ×ð’Û׋wslsdsUjÇ^zF hê -CJUVaJjhê -U hê -6]hê -h}#•56\]h56\] hhØ_Œj!ŠhhEHöÿUj³‡h†=*hEHòÿUj‹óœF hCJUVaJj4…h†=*hEHòÿUjYóœF hCJUVaJhjhUjƒhhEHöÿUjóœF hCJUVaJìIíIîI J!J-J.J1J3J7JbJgJuJvJzJ–J˜JµJ¿JÀJÁJÆJÈJÎJÓJÔJÕJÙJçJüJýJKKKK#K%K&K'K:K;K*\]hH56>*\]h‚#Þ56>*\]hÉ3Ï56>*\]h+4ÿhØ_ŒjoŠha#Ÿhê -EHòÿUjÍrzF hê -CJUVaJh¯¬ hê -6]j5€h4 šhê -EHúÿUjŽrzF hê -CJUVaJ h¥}\hê -jh¥}\hê -Uhê -!ÎNÐNöNøNúNüNþN O(O*OPOROTOVOXO†OˆO’O”OºOŒOŸOÀOÌOôOP P4P6PBPFPHPLPTPVP`PbP÷óä×÷óÍó÷óŸ±÷Íóªó÷ó›Ž÷óŠó†óª{pe^óªóª hØ_Œ5\hØ_Œ56>*\]h·II56>*\]h‚#Þ56>*\]hØ_Œhž"Cj~®hž_¹h‚#ÞEHöÿUjÇ^zF h‚#ÞCJUVaJ h‚#Þ6]j ¬h‚#Þh‚#ÞEHèÿUjSüœF h‚#ÞCJUVaJh‚#Þ56\]jÑšh†=*h‚#ÞEHòÿUj{[zF h‚#ÞCJUVaJh‚#Þjh‚#ÞU$bPšPªPÐPÒPÔPÖPØPèP.Q0QVQXQZQ\QfQhQŽQQ’Q”QRRRRRR?RARORüôüåØôÑÇÞ±¢•žÃž±†yžÃqÃbUqÃNà hê -6]j±žhì`…hê -EHŒÿUj€>zF hê -CJUVaJjhê -Uj†¶h/:Hhê -EHòÿUj¯izF hê -CJUVaJj[Žh/:Hhê -EHòÿUj“izF hê -CJUVaJ h¥}\hê -jh¥}\hê -Uhê -h‚#Þ56\] h‚#Þh‚#Þjœ°h‚#Þh‚#ÞEHâÿUjNýœF h‚#ÞCJUVaJjh‚#ÞUh‚#ÞRRPRhR¬RÄRÊRâRTT(TZT_TyTT—TšTÃTÇTßTUT?TBTCTVTùõîæõ×Êæõæõ»®æõæõŸ’æõæõƒvæõæõgZæõæõj²ØhŒ{Chê -EH˜ÿUj?FzF hê -CJUVaJjÏÔhk~hê -EHäÿUjïDzF hê -CJUVaJjÙÑhk~hê -EHäÿUjýCzF hê -CJUVaJjÒÏh†=*hê -EHòÿUjôBzF hê -CJUVaJjkÍhk~hê -EHäÿUjOCzF hê -CJUVaJjhê -U hk~hê -hê - hê -6]#VTWTXTYTaTbTuTvTwTxT~TT€T“T”T•T–T«T¬T¿TÀTÁTÂTÇTÈTÛTÜTÝTÞTüTýTUUðãÛ×Û×È»Û×·Ûך›Û×Û׌Û×Û×pcÛ×Û×Tj'KzF hê -CJUVaJjhîhÆYhê -EHàÿUjSJzF hê -CJUVaJjnëhÆYhê -EHòÿUjâIzF hê -CJUVaJjÞæhÆYhê -EH®ÿUjüHzF hê -CJUVaJhÉ3Ïjêâh†=*hê -EHàÿUj¬BzF hê -CJUVaJhê -jhê -UjWßhÆYhê -EHàÿUj–HzF hê -CJUVaJ UUU#U$U7U8U9U:UU?U–U¥UÅUëU VTVhV WFXyXY5YZZÈZ[:[v[_\ \óóóóóóóóóëãóóóó×ëÏÏÇÇÏÏdhgdÉ3ÏdhgdØ_Œ „hdà^„hgdØ_Œdàgdê -dàgdØ_Œ „hdh^„hgdØ_ŒÄWÆWÈWÊWXX.X/X0X1XIXJX]X^X_XaXtXuXvXwXxXyX¡X¢XµX¶X·XžXYYY0YðãÛ×Û×È»Û×Û׬ŸÛאƒÛ×|×Û×m`Û×\T\jhê -Uhê -jâ hÃËhŒA`EHöÿUj3ÿœF hŒA`CJUVaJ hŒA`hŒA`j hŒA`hŒA`EHöÿUjÿœF hŒA`CJUVaJjÖ h‚#ÞhŒA`EHâÿUjãþœF hŒA`CJUVaJj8 h‚#ÞhŒA`EHâÿUjNýœF hŒA`CJUVaJhŒA`jhŒA`Ujýh†=*hŒA`EHòÿUjõþœF hŒA`CJUVaJ0Y1Y2Y3Y5YUYZZZ(Z.Z0Z4Z|Z~ZŠZšZºZŒZÐZÒZäZæZ[[*[,[.[0[8[:[>[F[H[n[p[ðãÛ×ÓÏÓÅ·¬¡š×–׏׏׏׏×Û׀sÛ×o×oÛ×`jî8žF h·IICJUVaJhÉ3Ïj¢ h Dhê -EHèÿUjl{F hê -CJUVaJ hê -6]h\÷ hØ_Œ5\hØ_Œ56>*\]h\÷56>*\]hönühê -56>*\]hŒA`hŒA`6]hØ_ŒhŒA`hê -jhê -Uj hQc£hê -EHâÿUj±SzF hê -CJUVaJ#p[r[t[a\b\u\v\w\x\ˆ\‰\œ\\ž\Ÿ\¢\£\¶\·\ž\¹\Æ\Ç\Ú\Û\Ü\Ý\ ] ]] ]!]#]6]òêæêæ×Êêæêæ»®êæê柒êæêæƒvêæêægZêæj# h Dhê -EHöÿUjn{F hê -CJUVaJjÐ h Dhê -EHöÿUj]n{F hê -CJUVaJj€ h Dhê -EHöÿUjFn{F hê -CJUVaJj5 h Dhê -EHöÿUj n{F hê -CJUVaJjí h Dhê -EHöÿUjým{F hê -CJUVaJhê -jhê -Ujô hÉ3Ïh·IIEHúÿU! \Þ\þ]F^Ž^Ö^ï^5_M_j_©_`` aPaŒa‰bËb©cñc:d‚dÈdàdýde9eh÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷dhgdØ_Œ6]7]8]:]M]N]O]P]\]]]p]q]r]s]ƒ]„]—]˜]™]š]þ]ÿ]^^^^^^*^+^ðãÛ×È»Û×Û׬ŸÛ×ÛאƒÛ×Û×tgÛ×Û×XjÈo{F hê -CJUVaJj* h Dhê -EHöÿUj‰o{F hê -CJUVaJj°' h6Y~hê -EHöÿUja{{F hê -CJUVaJj_% h6Y~hê -EHöÿUjân{F hê -CJUVaJjS# h Dhê -EHöÿUj¿n{F hê -CJUVaJhê -jhê -Uj;! h Dhê -EHöÿUj©n{F hê -CJUVaJ+^,^-^.^/^B^C^D^E^F^G^Z^[^\^]^^^_^r^s^t^u^v^w^Š^‹^Œ^^Ž^^¢^£^€^¥^Š^§^º^òêæêæ×Êêæêæ»®êæê柒êæêæƒvêæêægZêæêæj68 h Dhê -EHöÿUj^p{F hê -CJUVaJjÜ5 h Dhê -EHöÿUj=p{F hê -CJUVaJj…3 h Dhê -EHöÿUj'p{F hê -CJUVaJj/1 h Dhê -EHöÿUjüo{F hê -CJUVaJjÑ. h Dhê -EHöÿUjæo{F hê -CJUVaJhê -jhê -Uj{, h Dhê -EHöÿU#º^»^Œ^œ^Ÿ^¿^Ò^Ó^Ô^Õ^Ö^×^ê^ë^ì^í^ _ _ _!_"_#_5_6_I_J_K_L_R_S_f_g_ðãÛ×Û×È»Û×Û׬ŸÛ×ÛאƒÛ×Û×tgÛ×Û×Xjr{F hê -CJUVaJj¬C hÄ3"hê -EHôÿUj@q{F hê -CJUVaJj A hÄ3"hê -EHôÿUjÿp{F hê -CJUVaJjG? 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The search for higher reciprocity laws eventually led to the development of a new field of mathematics known as algebraic number theory. References Beachy, John A., and William D. Blair. Abstract Algebra. Prospect Heights: Waveland Press Inc., 1996. Bell, E. T., Men of Mathematics. New York: Simon and Schuster, 1937. Buhler, W.K., Gauss: A Biographical Study. New York: Springer-Verlag, 1981. Burn, R. P., A Pathway into Number Theory. New York: Cambridge University Press, 1997. Burton, David M., Elementary Number Theory. Iowa: Wm. C. Brown Publishers, 1989. Cohn, Harvey, A second Course in Number Theory. New York: John Wiley & Sons, Inc., 1962. Dunham, William, Journey Through Genius. New York: Penguin Books, 1991. Goldman, Jay R., The Queen of Mathematics: A Historically Motivated Guide to Number Theory. Wellesly: A K Peters, 1998. Hall, Tord, Carl Friedrich Gauss. Trans. Albert Froderberg. Cambridge: MIT, 1970. Long, Calvin T., Elementary Introduction to Number Theory. New York: D.C. Heath and Company, 1965. Reich, Karin, Carl Friedrich Gauss. Trans. Patricia Crampton. Munich: Heinz Moos Verlag, 1977. Rousseau, G. On the Quadratic Reciprocity Law. Journal of Australian Mathematical Society (Series A). 51 (1991) 423-425. Schmidt, H. Drei neue Beweise des Reciprocitatssatzes in der Theorie der quadratischen Reste., Journal Reine Angew. Math. 111 (1893), 107-120. Schumer, Peter D., Introduction to Number Theory. Boston: PWS Publishing Company, 1996. Smith, David Eugene, A Source Book in Mathematics. New York: McGraw-Hill Book Company, Inc., 1929. “The Law of Quadratic Reciprocity.” http://www. Math.nott.ac.uk/personal/jec/courses/G13NUM/notes/node20.html. “The Jewel of Arithmetic: Quadratic Reciprocity.” http://www.mathpages.com/home/kmath075.htm. 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ˆ1†Å"ˆ2†Å"ˆ3†Å"†Å"†Å"‚(ƒp†"ˆ1‚)†Å"‚(ƒp†+ˆ1‚)‚(ƒp†+ˆ2‚)†Å"†Å"†Å"‚(ˆ2ƒp†"ˆ1‚)‚(ˆ2ƒp†+ˆ1‚)†Å"†Å"†Å"‚(ƒp†Å"ƒq†"ˆ1ˆ2†"ˆ1‚)†=ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)ObjInfoÿÿÿÿýÿÿÿÿ Equation Native ÿÿÿÿÿÿÿÿÿÿÿÿ _1182417409ÎÀFÐÈç,šÅÀJÊç,šÅOle ÿÿÿÿÿÿÿÿÿÿÿÿƒp†+ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)ƒq†"ˆ1ˆ2†"ˆ1–(–)ƒp†+ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)þÿ ÿÿÿÿÎÀFMicrosoft Equation 3.0 DS Equation Equation.3ô9²qCompObjÿÿÿÿÿfObjInfoÿÿÿÿÿÿÿÿEquation Native ÿÿÿÿÿÿÿÿÿÿÿÿg_1182417383ÿÿÿÿÿÿÿÿÎÀFÀJÊç,šÅÀJÊç,šÅÂKè£¬á ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)ƒp†+ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)†Å"†Å"†Å"†Å"ƒq†"ˆ1ˆ2†"ˆ1–(–)ƒp†+ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)†a"ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)†Å"†Å"†Å"†Å"ƒi ƒi†=ˆ1ƒp†"ˆ1 †" –(–)†a"‚(ƒp†"ˆ1‚)‚!‚(ƒp†"ˆ1‚)‚!†Å"†Å"†Å"‚(ƒp†"ˆ1‚)‚!‚(‚m‚o‚dƒp‚)þÿ ÿÿÿÿÎÀFMicrosoft Equation 3.0 DS Equation Equation.3ô9²qÂEpKÄß ‚(ƒp†"ˆ1‚)‚!†a"†"ˆ1‚(‚m‚o‚dƒp‚)Ole ÿÿÿÿÿÿÿÿÿÿÿÿ"CompObjÿÿÿÿ#fObjInfoÿÿÿÿÿÿÿÿ%Equation Native ÿÿÿÿÿÿÿÿÿÿÿÿ&a_1182417422ÿÿÿÿ  ÎÀFÀJÊç,šÅÀJÊç,šÅOle ÿÿÿÿÿÿÿÿÿÿÿÿ(CompObj  ÿÿÿÿ)fObjInfoÿÿÿÿ ÿÿÿÿ+þÿ ÿÿÿÿÎÀFMicrosoft Equation 3.0 DS 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