Math 412: Number Theory Lecture 25 Gaussian Integers

Transcription

1 Math 412: Number Theory Lecture 25 Gaussian Integers Gexin Yu College of William and Mary

2 Gaussian Integers and norm function Let i = p 1. Complex numbers of the form a + bi with a, b 2 Z are called Gaussian integers. The set is denoted Z[i]. We call a, b to be rational integers (di erent from Gaussian integers).

3 Gaussian Integers and norm function Let i = p 1. Complex numbers of the form a + bi with a, b 2 Z are called Gaussian integers. The set is denoted Z[i]. We call a, b to be rational integers (di erent from Gaussian integers). Thm: let, 2 Z[i]. Then +,, are all in Z[i].

4 Gaussian Integers and norm function Let i = p 1. Complex numbers of the form a + bi with a, b 2 Z are called Gaussian integers. The set is denoted Z[i]. We call a, b to be rational integers (di erent from Gaussian integers). Thm: let, 2 Z[i]. Then +,, are all in Z[i]. Let z = a + bi and define the norm of z to be N(z) = z 2 = zz = a 2 + b 2

5 Gaussian Integers and norm function Let i = p 1. Complex numbers of the form a + bi with a, b 2 Z are called Gaussian integers. The set is denoted Z[i]. We call a, b to be rational integers (di erent from Gaussian integers). Thm: let, 2 Z[i]. Then +,, are all in Z[i]. Let z = a + bi and define the norm of z to be N(z) = z 2 = zz = a 2 + b 2 The properties of the norm function N:

6 Gaussian Integers and norm function Let i = p 1. Complex numbers of the form a + bi with a, b 2 Z are called Gaussian integers. The set is denoted Z[i]. We call a, b to be rational integers (di erent from Gaussian integers). Thm: let, 2 Z[i]. Then +,, are all in Z[i]. Let z = a + bi and define the norm of z to be N(z) = z 2 = zz = a 2 + b 2 The properties of the norm function N: I (a) N(z) 0;

7 Gaussian Integers and norm function Let i = p 1. Complex numbers of the form a + bi with a, b 2 Z are called Gaussian integers. The set is denoted Z[i]. We call a, b to be rational integers (di erent from Gaussian integers). Thm: let, 2 Z[i]. Then +,, are all in Z[i]. Let z = a + bi and define the norm of z to be N(z) = z 2 = zz = a 2 + b 2 The properties of the norm function N: I (a) N(z) 0; I (b) N(zw) =N(z)N(w);

8 Gaussian Integers and norm function Let i = p 1. Complex numbers of the form a + bi with a, b 2 Z are called Gaussian integers. The set is denoted Z[i]. We call a, b to be rational integers (di erent from Gaussian integers). Thm: let, 2 Z[i]. Then +,, are all in Z[i]. Let z = a + bi and define the norm of z to be N(z) = z 2 = zz = a 2 + b 2 The properties of the norm function N: I (a) N(z) 0; I (b) N(zw) =N(z)N(w); I (c) N(z) =0i z = 0.

9 Divisibility and units Def: divides if = for some 2 Z[i]; and write as.

10 Divisibility and units Def: divides if = for some 2 Z[i]; and write as. Ex: 2 i 13 + i, but3+2i 6 6+5i.

11 Divisibility and units Def: divides if = for some 2 Z[i]; and write as. Ex: 2 i 13 + i, but3+2i 6 6+5i. Properties:

12 Divisibility and units Def: divides if = for some 2 Z[i]; and write as. Ex: 2 i 13 + i, but3+2i 6 6+5i. Properties: I (a) if z w, w v, thenz v;

13 Divisibility and units Def: divides if = for some 2 Z[i]; and write as. Ex: 2 i 13 + i, but3+2i 6 6+5i. Properties: I (a) if z w, w v, thenz v; I (b) if z u, z v, thenz xu + yv, wherez, w, u, v, x, y 2 Z[i].

14 Divisibility and units Def: divides if = for some 2 Z[i]; and write as. Ex: 2 i 13 + i, but3+2i 6 6+5i. Properties: I (a) if z w, w v, thenz v; I (b) if z u, z v, thenz xu + yv, wherez, w, u, v, x, y 2 Z[i]. Units: e is called a unit if e 1. When e is a unit, ea is an associate of the Gaussian integer a.

15 Divisibility and units Def: divides if = for some 2 Z[i]; and write as. Ex: 2 i 13 + i, but3+2i 6 6+5i. Properties: I (a) if z w, w v, thenz v; I (b) if z u, z v, thenz xu + yv, wherez, w, u, v, x, y 2 Z[i]. Units: e is called a unit if e 1. When e is a unit, ea is an associate of the Gaussian integer a. Thm: e 2 Z[i] isaunitif and only if N(e) = 1. It follows that the Gaussian integers that are units are 1, 1, i, i.

16 Division algorithm Thm: let and be Gaussian integers with 6= 0. Then there exist Gaussian integers and such that = +, and 0 apple N( ) < N( ).

17 Division algorithm Thm: let and be Gaussian integers with 6= 0. Then there exist Gaussian integers and such that = +, and 0 apple N( ) < N( ). Proof: suppose / = x + yi and let s =[x +0.5] and t =[y +0.5] (the closest integers to s, t, round up if x = y =0.5). Then we let = s + ti and =.ThenwecanshowthatN( ) < N( ).

18 Division algorithm Thm: let and be Gaussian integers with 6= 0. Then there exist Gaussian integers and such that = +, and 0 apple N( ) < N( ). Proof: suppose / = x + yi and let s =[x +0.5] and t =[y +0.5] (the closest integers to s, t, round up if x = y =0.5). Then we let = s + ti and =.ThenwecanshowthatN( ) < N( ). Remark: and are not unique any more. For example. = i, = 3+5i. Follow the steps in the above proof, we can get =2 4i and = 1 2i. But we can also choose =2 3i and =4+i.

19 GCD The GCD of two Gaussian integers and so that (a) and ; (b) if and,then. is the Gaussian integer

20 GCD The GCD of two Gaussian integers and so that (a) and ; (b) if and,then. is the Gaussian integer THM: if and are Gaussian integers, not both zero, then there exists a gcd of and ;alsothereexistsu, v 2 Z[i] (Bezout coe cients of and )suchthat = u + v.

21 GCD The GCD of two Gaussian integers and so that (a) and ; (b) if and,then. is the Gaussian integer THM: if and are Gaussian integers, not both zero, then there exists a gcd of and ;alsothereexistsu, v 2 Z[i] (Bezout coe cients of and )suchthat = u + v. Proof: Take the set S = {N(u + v ):u, v 2 Z[i]}. ThenS is nonempty and a subset of N. SoS has a least element, that is, some with = u 0 + v 0 such that N( ) is the least among all linear combinations. One can show that is the gcd, and divides every linear combination of and.

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