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Quadratic Reciprocity Quadratic Reciprocity Jennifer Li Department of Mathematics Louisiana State University Baton Rouge Theorem. Let n be a nonzero integer. Then there is a prime factorization Y n ( 1)²(n) pa(p) Æ ¡ p 0 È with the exponents uniquely determined by n. Note: ²(n) = 0 or 1 (depends on sign of n). More generally: k[x]: Ring of polynomials with coefficients in field k ² Even more generally: Principal Ideal Domains ² Unique Factorization Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 2 / 79 Note: ²(n) = 0 or 1 (depends on sign of n). More generally: k[x]: Ring of polynomials with coefficients in field k ² Even more generally: Principal Ideal Domains ² Unique Factorization Theorem. Let n be a nonzero integer. Then there is a prime factorization Y n ( 1)²(n) pa(p) Æ ¡ p 0 È with the exponents uniquely determined by n. Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 2 / 79 More generally: k[x]: Ring of polynomials with coefficients in field k ² Even more generally: Principal Ideal Domains ² Unique Factorization Theorem. Let n be a nonzero integer. Then there is a prime factorization Y n ( 1)²(n) pa(p) Æ ¡ p 0 È with the exponents uniquely determined by n. Note: ²(n) = 0 or 1 (depends on sign of n). Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 2 / 79 Even more generally: Principal Ideal Domains ² Unique Factorization Theorem. Let n be a nonzero integer. Then there is a prime factorization Y n ( 1)²(n) pa(p) Æ ¡ p 0 È with the exponents uniquely determined by n. Note: ²(n) = 0 or 1 (depends on sign of n). More generally: k[x]: Ring of polynomials with coefficients in field k ² Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 2 / 79 Unique Factorization Theorem. Let n be a nonzero integer. Then there is a prime factorization Y n ( 1)²(n) pa(p) Æ ¡ p 0 È with the exponents uniquely determined by n. Note: ²(n) = 0 or 1 (depends on sign of n). More generally: k[x]: Ring of polynomials with coefficients in field k ² Even more generally: Principal Ideal Domains ² Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 2 / 79 such that for all a,b R, with b 0, there exists c,d R where 2 6Æ 2 a cb d and Æ Å N(d) N(b) or d 0 Ç Æ Examples: Z and k[x] An integral domain R is a Principal Ideal Domain if every ideal of R is principal. Every Euclidean Domain is a Principal Ideal Domain Euclidean and Principal Ideal Domains An integral domain R is a Euclidean Domain if there is a normN N : R\{0} N ¡! Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 3 / 79 Examples: Z and k[x] An integral domain R is a Principal Ideal Domain if every ideal of R is principal. Every Euclidean Domain is a Principal Ideal Domain Euclidean and Principal Ideal Domains An integral domain R is a Euclidean Domain if there is a normN N : R\{0} N ¡! such that for all a,b R, with b 0, there exists c,d R where 2 6Æ 2 a cb d and Æ Å N(d) N(b) or d 0 Ç Æ Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 3 / 79 An integral domain R is a Principal Ideal Domain if every ideal of R is principal. Every Euclidean Domain is a Principal Ideal Domain Euclidean and Principal Ideal Domains An integral domain R is a Euclidean Domain if there is a normN N : R\{0} N ¡! such that for all a,b R, with b 0, there exists c,d R where 2 6Æ 2 a cb d and Æ Å N(d) N(b) or d 0 Ç Æ Examples: Z and k[x] Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 3 / 79 Every Euclidean Domain is a Principal Ideal Domain Euclidean and Principal Ideal Domains An integral domain R is a Euclidean Domain if there is a normN N : R\{0} N ¡! such that for all a,b R, with b 0, there exists c,d R where 2 6Æ 2 a cb d and Æ Å N(d) N(b) or d 0 Ç Æ Examples: Z and k[x] An integral domain R is a Principal Ideal Domain if every ideal of R is principal. Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 3 / 79 Euclidean and Principal Ideal Domains An integral domain R is a Euclidean Domain if there is a normN N : R\{0} N ¡! such that for all a,b R, with b 0, there exists c,d R where 2 6Æ 2 a cb d and Æ Å N(d) N(b) or d 0 Ç Æ Examples: Z and k[x] An integral domain R is a Principal Ideal Domain if every ideal of R is principal. Every Euclidean Domain is a Principal Ideal Domain Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 3 / 79 For ® a b! Z[!], define Æ Å 2 N(®) a2 ab b2 Æ ¡ Å Z[!] is a Euclidean Domain. ) Z[!] is also a Principal Ideal Domain. ) Example The ring Z[!] {a b! a,b Z} where ! ( 1 p 3)/2 Æ Å j 2 Æ ¡ Å ¡ Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 4 / 79 Z[!] is a Euclidean Domain. ) Z[!] is also a Principal Ideal Domain. ) Example The ring Z[!] {a b! a,b Z} where ! ( 1 p 3)/2 Æ Å j 2 Æ ¡ Å ¡ For ® a b! Z[!], define Æ Å 2 N(®) a2 ab b2 Æ ¡ Å Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 4 / 79 Z[!] is also a Principal Ideal Domain. ) Example The ring Z[!] {a b! a,b Z} where ! ( 1 p 3)/2 Æ Å j 2 Æ ¡ Å ¡ For ® a b! Z[!], define Æ Å 2 N(®) a2 ab b2 Æ ¡ Å Z[!] is a Euclidean Domain. ) Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 4 / 79 Example The ring Z[!] {a b! a,b Z} where ! ( 1 p 3)/2 Æ Å j 2 Æ ¡ Å ¡ For ® a b! Z[!], define Æ Å 2 N(®) a2 ab b2 Æ ¡ Å Z[!] is a Euclidean Domain. ) Z[!] is also a Principal Ideal Domain. ) Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 4 / 79 Caution B A prime in Z may not be a prime in Z[!] Example: p 7 is prime in Z. Æ But in Z[!]: 7 (3 !)(2 !) Æ Å ¡ In Z: “rational primes p" In Z[!]: “primes ¼" Primes in Z[!] Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 5 / 79 A prime in Z may not be a prime in Z[!] Example: p 7 is prime in Z. Æ But in Z[!]: 7 (3 !)(2 !) Æ Å ¡ In Z: “rational primes p" In Z[!]: “primes ¼" Primes in Z[!] Caution B Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 5 / 79 Example: p 7 is prime in Z. Æ But in Z[!]: 7 (3 !)(2 !) Æ Å ¡ In Z: “rational primes p" In Z[!]: “primes ¼" Primes in Z[!] Caution B A prime in Z may not be a prime in Z[!] Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 5 / 79 In Z: “rational primes p" In Z[!]: “primes ¼" Primes in Z[!] Caution B A prime in Z may not be a prime in Z[!] Example: p 7 is prime in Z. Æ But in Z[!]: 7 (3 !)(2 !) Æ Å ¡ Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 5 / 79 In Z[!]: “primes ¼" Primes in Z[!] Caution B A prime in Z may not be a prime in Z[!] Example: p 7 is prime in Z. Æ But in Z[!]: 7 (3 !)(2 !) Æ Å ¡ In Z: “rational primes p" Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 5 / 79 Primes in Z[!] Caution B A prime in Z may not be a prime in Z[!] Example: p 7 is prime in Z. Æ But in Z[!]: 7 (3 !)(2 !) Æ Å ¡ In Z: “rational primes p" In Z[!]: “primes ¼" Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 5 / 79 Then a b(mod m) means that m divides b a. ´ ¡ Congruence Suppose a,b,m Z and m 0 2 6Æ Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 6 / 79 Congruence Suppose a,b,m Z and m 0 2 6Æ Then a b(mod m) means that m divides b a. ´ ¡ Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 6 / 79 xn a(mod p) ´ Fp: Finite field with p elements. Finite Fields Let a,n Z. 2 Suppose p is prime and and gcd (a,p) 1. Æ Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 7 / 79 Fp: Finite field with p elements. Finite Fields Let a,n Z. 2 Suppose p is prime and and gcd (a,p) 1. Æ xn a(mod p) ´ Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 7 / 79 Finite Fields Let a,n Z. 2 Suppose p is prime and and gcd (a,p) 1. Æ xn a(mod p) ´ Fp: Finite field with p elements. Jennifer Li (Louisiana State University) Quadratic Reciprocity May 8, 2015 7 / 79 Elements: roots of xq x. ¡ Multiplicative group of F: denoted Fq¤.
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