Euclidean Domains

Euclidean Domains

Euclidean Domains Kevin James Kevin James Euclidean Domains Definition An integral domain R is called a Euclidean Domain if R has a division algorithm. That is, if there is a norm N of R such that for any a; b 2 R with b 6= 0R there exists q; r 2 R satisfying: 1 a = bq + r, and 2 r = 0R or N(r) < N(b). For such q; r 2 R, q is called a quotient and r is called a remainder upon divison of a by b. Definition Suppose that R is an integral domain. Any function N : R ! N [ f0g with N(0) = 0 is a norm. If N(a) > 0, 8a 2 R n f0R g, then N is called a positive norm. Kevin James Euclidean Domains Definition Suppose that R is an integral domain. Any function N : R ! N [ f0g with N(0) = 0 is a norm. If N(a) > 0, 8a 2 R n f0R g, then N is called a positive norm. Definition An integral domain R is called a Euclidean Domain if R has a division algorithm. That is, if there is a norm N of R such that for any a; b 2 R with b 6= 0R there exists q; r 2 R satisfying: 1 a = bq + r, and 2 r = 0R or N(r) < N(b). For such q; r 2 R, q is called a quotient and r is called a remainder upon divison of a by b. Kevin James Euclidean Domains Proposition If R is a Euclidean Domain, then every ideal is principal. More precisely, if I E R , then I = (d) where d is any element of I of minimum norm. Example • Z is a Euclidean domain and thus all ideals of Z are principal. • Q[x] is a Euclidean domain. • Z[x] is not a Euclidean domain since one can check that (3; x) is not principal. Note The existence of a division algorithm for a ring R allows employment of the Euclidean Algorithm in R for computation of greatest common divisors. Kevin James Euclidean Domains Example • Z is a Euclidean domain and thus all ideals of Z are principal. • Q[x] is a Euclidean domain. • Z[x] is not a Euclidean domain since one can check that (3; x) is not principal. Note The existence of a division algorithm for a ring R allows employment of the Euclidean Algorithm in R for computation of greatest common divisors. Proposition If R is a Euclidean Domain, then every ideal is principal. More precisely, if I E R , then I = (d) where d is any element of I of minimum norm. Kevin James Euclidean Domains Note The existence of a division algorithm for a ring R allows employment of the Euclidean Algorithm in R for computation of greatest common divisors. Proposition If R is a Euclidean Domain, then every ideal is principal. More precisely, if I E R , then I = (d) where d is any element of I of minimum norm. Example • Z is a Euclidean domain and thus all ideals of Z are principal. • Q[x] is a Euclidean domain. • Z[x] is not a Euclidean domain since one can check that (3; x) is not principal. Kevin James Euclidean Domains Proposition If a; b 2 R and (a; b) = (d) then d is a greatest common divisor of a and b. Proposition Suppose that R is an integral domain and that d; d0 2 R. If (d) = (d0), then there is a unit u 2 R× such that d = ud0. In particular, if d and d0 are greatest common divisors of a and b then d = ud0 for some u 2 R×. Definition Let R be a commutative ring and let a; b 2 R with b 6= 0R . 1 We say that a is a multiple of b if there is c 2 R such that a = bc. We also say that b divides a and write bja. 2 A greatest common divisor of a and b (if it exists) is an element d 2 R satisfying 1 dja and djb, and 2 if d 0 2 R, d 0ja and d 0jb then d 0jd also. Kevin James Euclidean Domains Proposition Suppose that R is an integral domain and that d; d0 2 R. If (d) = (d0), then there is a unit u 2 R× such that d = ud0. In particular, if d and d0 are greatest common divisors of a and b then d = ud0 for some u 2 R×. Definition Let R be a commutative ring and let a; b 2 R with b 6= 0R . 1 We say that a is a multiple of b if there is c 2 R such that a = bc. We also say that b divides a and write bja. 2 A greatest common divisor of a and b (if it exists) is an element d 2 R satisfying 1 dja and djb, and 2 if d 0 2 R, d 0ja and d 0jb then d 0jd also. Proposition If a; b 2 R and (a; b) = (d) then d is a greatest common divisor of a and b. Kevin James Euclidean Domains Definition Let R be a commutative ring and let a; b 2 R with b 6= 0R . 1 We say that a is a multiple of b if there is c 2 R such that a = bc. We also say that b divides a and write bja. 2 A greatest common divisor of a and b (if it exists) is an element d 2 R satisfying 1 dja and djb, and 2 if d 0 2 R, d 0ja and d 0jb then d 0jd also. Proposition If a; b 2 R and (a; b) = (d) then d is a greatest common divisor of a and b. Proposition Suppose that R is an integral domain and that d; d0 2 R. If (d) = (d0), then there is a unit u 2 R× such that d = ud0. In particular, if d and d0 are greatest common divisors of a and b then d = ud0 for some u 2 R×. Kevin James Euclidean Domains Theorem Let R be a Euclidean domain and let a; b 2 R be non-zero. Let d = rn be the final non-zero remainder in the Euclidean Algorithm. 1 d is a greatest common divisor of a and b, and 2 (a; b) = (d). In particular, d is an R-linear combination of a and b. That is there are x; y 2 R such that d = ax + by. Kevin James Euclidean Domains Proposition Let R be an integral domain that is not a field. If R is a Euclidean domain then R has universal side divisors. Example p (1+ −19) R = Z 2 is an integral domain which has no universal side divisors and is therefore not a Euclidean domain. Definition × Suppose that R is an integral domain. Denote by R~ = R [ f0R g. We say that u 2 R n R~ is a universal side divisor if 8x 2 R, 9z 2 R~ such that u divides x − z. That is, there is q 2 R and z 2 R~ such that x = qu + z. Kevin James Euclidean Domains Example p (1+ −19) R = Z 2 is an integral domain which has no universal side divisors and is therefore not a Euclidean domain. Definition × Suppose that R is an integral domain. Denote by R~ = R [ f0R g. We say that u 2 R n R~ is a universal side divisor if 8x 2 R, 9z 2 R~ such that u divides x − z. That is, there is q 2 R and z 2 R~ such that x = qu + z. Proposition Let R be an integral domain that is not a field. If R is a Euclidean domain then R has universal side divisors. Kevin James Euclidean Domains Definition × Suppose that R is an integral domain. Denote by R~ = R [ f0R g. We say that u 2 R n R~ is a universal side divisor if 8x 2 R, 9z 2 R~ such that u divides x − z. That is, there is q 2 R and z 2 R~ such that x = qu + z. Proposition Let R be an integral domain that is not a field. If R is a Euclidean domain then R has universal side divisors. Example p (1+ −19) R = Z 2 is an integral domain which has no universal side divisors and is therefore not a Euclidean domain. Kevin James Euclidean Domains.

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