Section 46 – Euclidean domains Instructor: Yifan Yang Spring 2007 Instructor: Yifan Yang Section 46 – Euclidean domains
Division algorithms for Z and F [ x ] Theorem Let a and b be two integers with b � = 0 . Then there exist two integers q and r such that • a = bq + r, and • either r = 0 or | r | < | b | . Theorem (23.1) Let F be a field. Then for any a ( x ) , b ( x ) ∈ F [ x ] with b ( x ) � = 0 , there exist two polynomials q ( x ) and r ( x ) such that • a ( x ) = b ( x ) q ( x ) + r ( x ) , and • either r ( x ) = 0 or deg r ( x ) < deg b ( x ) . Instructor: Yifan Yang Section 46 – Euclidean domains
Division algorithms for Z and F [ x ] Theorem Let a and b be two integers with b � = 0 . Then there exist two integers q and r such that • a = bq + r, and • either r = 0 or | r | < | b | . Theorem (23.1) Let F be a field. Then for any a ( x ) , b ( x ) ∈ F [ x ] with b ( x ) � = 0 , there exist two polynomials q ( x ) and r ( x ) such that • a ( x ) = b ( x ) q ( x ) + r ( x ) , and • either r ( x ) = 0 or deg r ( x ) < deg b ( x ) . Instructor: Yifan Yang Section 46 – Euclidean domains
Division algorithms for Z and F [ x ] Theorem Let a and b be two integers with b � = 0 . Then there exist two integers q and r such that • a = bq + r, and • either r = 0 or | r | < | b | . Theorem (23.1) Let F be a field. Then for any a ( x ) , b ( x ) ∈ F [ x ] with b ( x ) � = 0 , there exist two polynomials q ( x ) and r ( x ) such that • a ( x ) = b ( x ) q ( x ) + r ( x ) , and • either r ( x ) = 0 or deg r ( x ) < deg b ( x ) . Instructor: Yifan Yang Section 46 – Euclidean domains
Division algorithms for Z and F [ x ] Theorem Let a and b be two integers with b � = 0 . Then there exist two integers q and r such that • a = bq + r, and • either r = 0 or | r | < | b | . Theorem (23.1) Let F be a field. Then for any a ( x ) , b ( x ) ∈ F [ x ] with b ( x ) � = 0 , there exist two polynomials q ( x ) and r ( x ) such that • a ( x ) = b ( x ) q ( x ) + r ( x ) , and • either r ( x ) = 0 or deg r ( x ) < deg b ( x ) . Instructor: Yifan Yang Section 46 – Euclidean domains
Euclidean algorithm Definition A Euclidean norm on an integral domain D is a function ν : D − { 0 } → N ∪ { 0 } such that the following conditions are satisfied: • For all a , b ∈ D with b � = 0, there exist q and r in D such that • a = bq + r , and • either r = 0 or ν ( r ) < ν ( b ) . • For all a , b ∈ D , where neither a nor b is 0, ν ( a ) ≤ ν ( ab ) . An integral domain D is a Euclidean domain if there exists a Euclidean norm on D . Instructor: Yifan Yang Section 46 – Euclidean domains
Euclidean algorithm Definition A Euclidean norm on an integral domain D is a function ν : D − { 0 } → N ∪ { 0 } such that the following conditions are satisfied: • For all a , b ∈ D with b � = 0, there exist q and r in D such that • a = bq + r , and • either r = 0 or ν ( r ) < ν ( b ) . • For all a , b ∈ D , where neither a nor b is 0, ν ( a ) ≤ ν ( ab ) . An integral domain D is a Euclidean domain if there exists a Euclidean norm on D . Instructor: Yifan Yang Section 46 – Euclidean domains
Euclidean algorithm Definition A Euclidean norm on an integral domain D is a function ν : D − { 0 } → N ∪ { 0 } such that the following conditions are satisfied: • For all a , b ∈ D with b � = 0, there exist q and r in D such that • a = bq + r , and • either r = 0 or ν ( r ) < ν ( b ) . • For all a , b ∈ D , where neither a nor b is 0, ν ( a ) ≤ ν ( ab ) . An integral domain D is a Euclidean domain if there exists a Euclidean norm on D . Instructor: Yifan Yang Section 46 – Euclidean domains
Euclidean algorithm Definition A Euclidean norm on an integral domain D is a function ν : D − { 0 } → N ∪ { 0 } such that the following conditions are satisfied: • For all a , b ∈ D with b � = 0, there exist q and r in D such that • a = bq + r , and • either r = 0 or ν ( r ) < ν ( b ) . • For all a , b ∈ D , where neither a nor b is 0, ν ( a ) ≤ ν ( ab ) . An integral domain D is a Euclidean domain if there exists a Euclidean norm on D . Instructor: Yifan Yang Section 46 – Euclidean domains
Euclidean algorithm Definition A Euclidean norm on an integral domain D is a function ν : D − { 0 } → N ∪ { 0 } such that the following conditions are satisfied: • For all a , b ∈ D with b � = 0, there exist q and r in D such that • a = bq + r , and • either r = 0 or ν ( r ) < ν ( b ) . • For all a , b ∈ D , where neither a nor b is 0, ν ( a ) ≤ ν ( ab ) . An integral domain D is a Euclidean domain if there exists a Euclidean norm on D . Instructor: Yifan Yang Section 46 – Euclidean domains
Examples • Define ν : Z → N ∪ { 0 } by ν ( n ) = | n | . Then ν is a Euclidean norm. • Define ν : F [ x ] − { 0 } → N ∪ { 0 } by ν ( f ( x )) = deg f ( x ) . Then ν is a Euclidean norm. Instructor: Yifan Yang Section 46 – Euclidean domains
Examples • Define ν : Z → N ∪ { 0 } by ν ( n ) = | n | . Then ν is a Euclidean norm. • Define ν : F [ x ] − { 0 } → N ∪ { 0 } by ν ( f ( x )) = deg f ( x ) . Then ν is a Euclidean norm. Instructor: Yifan Yang Section 46 – Euclidean domains
ED ⇒ PID Theorem (46.4) Every Euclidean domain D is a principal ideal domain. Corollary (46.5) Every Euclidean domain is a unique factorization domain. Proof. • Given an ideal I , if I = { 0 } , then I = � 0 � and we are done. • If I � = { 0 } , let b be an element of I of minimal norm. (That is, ν ( b ) = min a � = 0 ∈ I ν ( a ) .) We claim that I = � b � . • Given a ∈ I , there exist q and r in D such that a = bq + r with r = 0 or ν ( r ) < ν ( b ) . • The possibility ν ( r ) < ν ( b ) can not occur. Thus a = bq ∈ � b � . � Instructor: Yifan Yang Section 46 – Euclidean domains
ED ⇒ PID Theorem (46.4) Every Euclidean domain D is a principal ideal domain. Corollary (46.5) Every Euclidean domain is a unique factorization domain. Proof. • Given an ideal I , if I = { 0 } , then I = � 0 � and we are done. • If I � = { 0 } , let b be an element of I of minimal norm. (That is, ν ( b ) = min a � = 0 ∈ I ν ( a ) .) We claim that I = � b � . • Given a ∈ I , there exist q and r in D such that a = bq + r with r = 0 or ν ( r ) < ν ( b ) . • The possibility ν ( r ) < ν ( b ) can not occur. Thus a = bq ∈ � b � . � Instructor: Yifan Yang Section 46 – Euclidean domains
ED ⇒ PID Theorem (46.4) Every Euclidean domain D is a principal ideal domain. Corollary (46.5) Every Euclidean domain is a unique factorization domain. Proof. • Given an ideal I , if I = { 0 } , then I = � 0 � and we are done. • If I � = { 0 } , let b be an element of I of minimal norm. (That is, ν ( b ) = min a � = 0 ∈ I ν ( a ) .) We claim that I = � b � . • Given a ∈ I , there exist q and r in D such that a = bq + r with r = 0 or ν ( r ) < ν ( b ) . • The possibility ν ( r ) < ν ( b ) can not occur. Thus a = bq ∈ � b � . � Instructor: Yifan Yang Section 46 – Euclidean domains
ED ⇒ PID Theorem (46.4) Every Euclidean domain D is a principal ideal domain. Corollary (46.5) Every Euclidean domain is a unique factorization domain. Proof. • Given an ideal I , if I = { 0 } , then I = � 0 � and we are done. • If I � = { 0 } , let b be an element of I of minimal norm. (That is, ν ( b ) = min a � = 0 ∈ I ν ( a ) .) We claim that I = � b � . • Given a ∈ I , there exist q and r in D such that a = bq + r with r = 0 or ν ( r ) < ν ( b ) . • The possibility ν ( r ) < ν ( b ) can not occur. Thus a = bq ∈ � b � . � Instructor: Yifan Yang Section 46 – Euclidean domains
ED ⇒ PID Theorem (46.4) Every Euclidean domain D is a principal ideal domain. Corollary (46.5) Every Euclidean domain is a unique factorization domain. Proof. • Given an ideal I , if I = { 0 } , then I = � 0 � and we are done. • If I � = { 0 } , let b be an element of I of minimal norm. (That is, ν ( b ) = min a � = 0 ∈ I ν ( a ) .) We claim that I = � b � . • Given a ∈ I , there exist q and r in D such that a = bq + r with r = 0 or ν ( r ) < ν ( b ) . • The possibility ν ( r ) < ν ( b ) can not occur. Thus a = bq ∈ � b � . � Instructor: Yifan Yang Section 46 – Euclidean domains
ED ⇒ PID Theorem (46.4) Every Euclidean domain D is a principal ideal domain. Corollary (46.5) Every Euclidean domain is a unique factorization domain. Proof. • Given an ideal I , if I = { 0 } , then I = � 0 � and we are done. • If I � = { 0 } , let b be an element of I of minimal norm. (That is, ν ( b ) = min a � = 0 ∈ I ν ( a ) .) We claim that I = � b � . • Given a ∈ I , there exist q and r in D such that a = bq + r with r = 0 or ν ( r ) < ν ( b ) . • The possibility ν ( r ) < ν ( b ) can not occur. Thus a = bq ∈ � b � . � Instructor: Yifan Yang Section 46 – Euclidean domains
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