Math 412: Number Theory Lecture 26 Gaussian Integers II

Transcription

1 Math 412: Number Theory Lecture 26 Gaussian Integers II Gexin Yu College of William and Mary

2 Let i = 1. Complex numbers of the form a + bi with a, b Z are called Gaussian integers. Let z = a + bi and define the norm N(z) = z 2 = zz = a 2 + b 2 Def: α divides β if β = αγ for some γ Z[i]; and write as α β. Units: e is called a unit if e 1. The units are 1, 1, i, i. When e is a unit, ea is an associate of the Gaussian integer a. Thm: let α and β be Gaussian integers with β 0. Then there exist γ, ρ Z[i] such that α = βγ + ρ, and 0 N(ρ) < N(β). The GCD of two Gaussian integers α and β is the Gaussian integer γ so that (a) γ α and γ β; (b) if δ α and δ β, then δ γ. THM: if α and β are Gaussian integers, not both zero, then there exists a gcd γ of α and β; also there exists u, v Z[i] (Bezout coefficients of α and β) such that γ = uα + vβ.

3 GCD THM: if γ 1 and γ 2 are both gcd of α and β, not both zero, then γ 1 and γ 2 are associates of each other.

4 GCD THM: if γ 1 and γ 2 are both gcd of α and β, not both zero, then γ 1 and γ 2 are associates of each other. Def: α and β are coprime if their gcd are 1, 1, i, i.

5 GCD THM: if γ 1 and γ 2 are both gcd of α and β, not both zero, then γ 1 and γ 2 are associates of each other. Def: α and β are coprime if their gcd are 1, 1, i, i. Euclidean Algorithm: one can do the algorithm so that ρ i = ρ i+1 γ i+1 + r i+2 so that when N(ρ j ) = 0, ρ j 1 is the gcd.

6 Ex: Find the gcd of i and 6 + 5i.

7 Gaussian Prime A nonzero Gaussian integer π is a Gaussian prime if it is not a unit and is divisible only by units and its associates.

8 Gaussian Prime A nonzero Gaussian integer π is a Gaussian prime if it is not a unit and is divisible only by units and its associates. THM: if π is a Gaussian integer and N(π) = p, where p is a rational prime, then π and π are Gaussian primes, and p is not a Gaussian prime.

9 Gaussian Prime A nonzero Gaussian integer π is a Gaussian prime if it is not a unit and is divisible only by units and its associates. THM: if π is a Gaussian integer and N(π) = p, where p is a rational prime, then π and π are Gaussian primes, and p is not a Gaussian prime. Ex: 2 i is a Gaussian prime, and 5 is not a Gaussian prime; 2 + 3i is a Gaussian prime, and 13 is not a Gaussian prime.

10 Gaussian Prime A nonzero Gaussian integer π is a Gaussian prime if it is not a unit and is divisible only by units and its associates. THM: if π is a Gaussian integer and N(π) = p, where p is a rational prime, then π and π are Gaussian primes, and p is not a Gaussian prime. Ex: 2 i is a Gaussian prime, and 5 is not a Gaussian prime; 2 + 3i is a Gaussian prime, and 13 is not a Gaussian prime. Ex: 3 is a Gaussian prime, but N(3) = 9 is not a rational prime.

11 Gaussian Prime A nonzero Gaussian integer π is a Gaussian prime if it is not a unit and is divisible only by units and its associates. THM: if π is a Gaussian integer and N(π) = p, where p is a rational prime, then π and π are Gaussian primes, and p is not a Gaussian prime. Ex: 2 i is a Gaussian prime, and 5 is not a Gaussian prime; 2 + 3i is a Gaussian prime, and 13 is not a Gaussian prime. Ex: 3 is a Gaussian prime, but N(3) = 9 is not a rational prime. Ex: 2 is not a Gaussian prime, since 2 = (1 + i)(1 i), but 1 + i and 1 i are both prime.

12 Unique factorization THM: if π is a Gaussian prime and α and β are Gaussian integers such that π αβ, then π α or π β.

13 Unique factorization THM: if π is a Gaussian prime and α and β are Gaussian integers such that π αβ, then π α or π β. THM: suppose that γ is a nonzero Gaussian integer that is not a unit, then γ can be written as a product of Gaussian primes and up to associates, the factorization is unique.

14 Unique factorization THM: if π is a Gaussian prime and α and β are Gaussian integers such that π αβ, then π α or π β. THM: suppose that γ is a nonzero Gaussian integer that is not a unit, then γ can be written as a product of Gaussian primes and up to associates, the factorization is unique. Ex: 20 = (1 + i) 4 (1 + 2i)(1 2i).

15 Fermat s two square theorem Revisit THM: every prime of the form 4k + 1 can be written as a sum of two perfect squares, and the sum is unique.

16 Fermat s two square theorem Revisit THM: every prime of the form 4k + 1 can be written as a sum of two perfect squares, and the sum is unique. Pf: We observe that t 2 1 (mod p) has a solution, since 1 is quadratic residue modulo p. So p (t 2 + 1) = (t + i)(t i). But p t + i and p t i, so p is not a Gaussian prime. Let p = αβ. Then N(p) = N(αβ) = N(α)N(β) and thus p = N(α) = N(β).

17 Fermat s two square theorem Revisit THM: every prime of the form 4k + 1 can be written as a sum of two perfect squares, and the sum is unique. Pf: We observe that t 2 1 (mod p) has a solution, since 1 is quadratic residue modulo p. So p (t 2 + 1) = (t + i)(t i). But p t + i and p t i, so p is not a Gaussian prime. Let p = αβ. Then N(p) = N(αβ) = N(α)N(β) and thus p = N(α) = N(β). Lem: Let π be a Gaussian prime, then there is exactly one rational prime p such that π p.

18 Fermat s two square theorem Revisit THM: every prime of the form 4k + 1 can be written as a sum of two perfect squares, and the sum is unique. Pf: We observe that t 2 1 (mod p) has a solution, since 1 is quadratic residue modulo p. So p (t 2 + 1) = (t + i)(t i). But p t + i and p t i, so p is not a Gaussian prime. Let p = αβ. Then N(p) = N(αβ) = N(α)N(β) and thus p = N(α) = N(β). Lem: Let π be a Gaussian prime, then there is exactly one rational prime p such that π p. pf: π π = N(π) = p 1 p 2..p k, so π p i for some i. If π p 1 and π p 2, then π mp 1 + np 2 = 1, so π is a unit.

19 THM: let p be a rational prime. Then

20 THM: let p be a rational prime. Then (a) if p = 2, then p = (1 + i)(1 i) = i(1 i) 2 ;

21 THM: let p be a rational prime. Then (a) if p = 2, then p = (1 + i)(1 i) = i(1 i) 2 ; (b) if p 3 (mod 4), then p is a Gaussian prime.

22 THM: let p be a rational prime. Then (a) if p = 2, then p = (1 + i)(1 i) = i(1 i) 2 ; (b) if p 3 (mod 4), then p is a Gaussian prime. (c) if p 1 (mod 4), then p = ππ, where π and π are Gaussian primes that are not associates with N(π) = N(π ) = p.

23 THM: let p be a rational prime. Then (a) if p = 2, then p = (1 + i)(1 i) = i(1 i) 2 ; (b) if p 3 (mod 4), then p is a Gaussian prime. (c) if p 1 (mod 4), then p = ππ, where π and π are Gaussian primes that are not associates with N(π) = N(π ) = p. THM: if n = 2 m p e pe k k qf 1 1..qt ft with m 0, f i even, and p i of form 4t + 1, and q i of form 4t + 3, then n has 4(e 1 + 1)(e 2 + 1)...(e k + 1) ways to write as a sum of two squares.

24 Pythagorean triples: another application of Gaussian integers z 2 = x 2 + y 2 = (x + yi)(x yi):

25 Pythagorean triples: another application of Gaussian integers z 2 = x 2 + y 2 = (x + yi)(x yi): Claim: if (x, y) = 1 as rational integers, then x + yi and x yi are coprime as Gaussian integers.

26 Pythagorean triples: another application of Gaussian integers z 2 = x 2 + y 2 = (x + yi)(x yi): Claim: if (x, y) = 1 as rational integers, then x + yi and x yi are coprime as Gaussian integers. Lemma: if αβ = γ 2 and α and β are coprime, then α = uδ1 2 and β = δ2 2 /u, where u is a unit. (hw)

27 Pythagorean triples: another application of Gaussian integers z 2 = x 2 + y 2 = (x + yi)(x yi): Claim: if (x, y) = 1 as rational integers, then x + yi and x yi are coprime as Gaussian integers. Lemma: if αβ = γ 2 and α and β are coprime, then α = uδ1 2 and β = δ2 2 /u, where u is a unit. (hw) Therefore we have x + yi = (s + ti) 2, (s + ti) 2, i(s + ti) 2, i(s + ti) 2.