L24-16-Mar-FINAL.pdf - MATH 228 Lecture 24 16 March 2018...

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MATH 228 Lecture 24 16 March 2018
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Euclidean Domains L24.1. Definition. Let R be an integral domain. Then R is a Euclidean domain if there is a function δ : R \ { 0 } → { 0 , 1 , 2 , 3 , . . . } with the following two properties: (a) Whenever a, b R are non-zero, we have δ ( ab ) δ ( a ) , and δ ( b ). (b) If a, b R and b 6 = 0, then either b | a , or there exist elements q, r R such that a = bq + r and δ ( r ) < δ ( b ). L24.2. Example. The ring Z is a Euclidean domain, with d ( x ) = | x | for all non-zero x .
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More examples of Euclidean Domains L24.3. Example. If F is any field, then the ring of polynomials F [ x ], consisting of all polynomials in a variable x with coefficients in F , is a Euclidean domain with δ ( f ) = deg ( f ) for all non-zero polynomials f . L24.4. Example. The ring R = Z [ i ] of Gaussian integers (recall that i = - 1) is a Euclidean domain, with the following function δ : If a = x + iy , then δ ( a ) = x 2
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