NOTES ON MATH 571 (ABSTRACT ALGEBRA), CHAPTER VIII: EUCLIDEAN DOMAINS, PRINCIPAL IDEAL DOMAINS, UNIQUE FACTORIZATION DOMAINS LI LI We will talk about three special cases of commutative rings: fEuclidean Domainsg ⊂ fPrinciple Ideal Domainsg ⊂ fUnique Factoriza- tion Domainsg. They are all integral domains: Definition 1. An integral domain is a commutative ring with 1 6= 0 and has no zero divisor, i.e. if ab = 0 then either a = 0 or b = 0. Example. example includes Z, Z=pZ, Q[x]. Non-example: Z=4Z. + Definition 2. (1) A function N : R ! Z [ f0g with N(0) = 0 is called a norm on R. If N(a) => 0 we call N a positive norm. (2) The integral domain R is a Euclidean Domain if there is a norm N on R such that for any two elements a and b of R with b 6= 0 there exists elements q and r with a = qb + r; with r = 0 or N(r) < N(b). i 2 2 Example. Z, R[x]. Z[i] (Define the norm of a + b to be a + b . To prove (2), let a = x + yi, b = u + vi, a=b = r + si, take integer p to be closest to r and q to be closest to s. Let r = a − (p + qi)b. Then N(r) < N(b).) Proposition 3. Every ideal in a Euclidean Domain is principal. More precisely, if I is a nonzero ideal, then I = (d) where d is any nonzero element of minimal norm. Example.p Z[x] is not an Euclidean Domain. Consider the ideal (2;p x). Z[ −5] is not an Euclidean Domain. Consider the ideal (3; 2 + −5). Definition 4. Let R be a commutative ring and a; b 2 R with b 6= 0. (a) a is a multiple of b if there is x 2 R such that a = bx, in this case we say b divides a, written bja. (2) The greatest common divisor of a and b, denoted by (a; b), is a nonzero element d such that (i) dja and djb. (ii) if d0ja and d0jb, then d0jd. Example.p The greatest commonp divisor may not exist, for example, 9 and 3(2 + −5) in the ring Z[ −5]. 1 2 LI LI Proposition 5. If a and b are nonzero elements in the commutative ring R such that the ideal (a; b) = (d), then d is a greatest common divisor of a and b. How about the \uniqueness" of the generator of a principal ideal? Proposition 6. Let R be an integral domain and (d) = (d0), then d0 = ud for some unit u 2 R. In particular, if d and d0 are both greatest common divisor of a and b, then d0 = ud for some unit u. Is there any algorithm to compute the greatest common divisor? The answer is yes for a Euclidean domain, just like what we do for integers. Theorem 7. Let R be a Euclidean domain and a, b be nonzero elements. Let a = q0b + r0; b = q1r0 + r1; r0 = q2r1 + r2; : : : ; rn−1 = qn+1rn: Let d = rn. Then (1) d is the greatest common divisor of a and b, and (2) the principal ideal (d) = (a; b). In particular, d is an R-linear combi- nation of a; b, i.e., d = ax + by for some x; y 2 R. (For (1) Backward induction shows dja and djb; If d0ja and d0jb, forward 0 induction shows d jd. For (2), forward induction shows that each ri is an R-linear combination of a; b.) Example. Write the gcd of a = 69 and b = 372 as a linear combination of a; b. Example. Find a gcd for 85 and 1 + 13i in the Euclidean domain Z[i]. Example. A public Key code: let M; N; d be positive integers, (M; N) = 1, (d; '(N)) = 1. Here '(n) is the Euler's '-function, which is the order of the × group (Z=nZ) . d d0 0 Prove that if M1 = M (mod N), then M = M1 (modN) where dd ≡ 1 (mod '(N)). (How it works: make N, d public known and hide the factors of N. Everyone can encrypt a message with the public key, but it can only be decrypted by the receiver who has the private key, like a mailbox with a mail slot.) CHAPTER VIII 3 Principal Ideal Domains Definition 8. A Principal Ideal Domain is an integral domain in which every ideal is principal. We have shown that every Euclidean Domain is a Principal Ideal Domain. But not otherwise. p p The ring ofp integers Z[!] of Q( D), where ! = D if D ≡ 2; 3 mod 4 and ! = (1 + D)=2 if D ≡ 1 mod 4, are P.I.D. for D = −1; −2; −3; −7; −11; −19; −43; −67; −163: But among them, for D = −1; −2; −3; −7; −11, Z[!] are Euclidean and others are not. Some facts on the greatest common divisor: Proposition 9. Let R be a P.I.D, a; b be two nonzero elements in R and (d) = (a; b). Then (1) d is a greatest common divisor of a and b. (2) There exist x; y 2 R such that d = ax + by. (3) d is unique up to multiplication by a unit in R. (Has been proved in previous class.) Unique Factorization Domains Definition 10. Let R be an integral domain. (1) Let r 2 R be nonzero and nonunit. Then r is called irreducible in R if whenever r = ab, at least one of a or b must be a unit. Otherwise r is called reducible. (2) The nonzero element p 2 R is called a prime in R if the ideal (p) is a prime ideal, i.e. if pjab then pja or pjb. Example. Z, Z[x]. Proposition 11. In an integral domain a prime element is always irre- ducible. Proposition 12. In a P.I.D, a nonzero element is a prime iff it is irre- ducible. p Remark. But in general, prime is a stronger notion. For example, in Z[ −5], 3 is irreducible but not prime. Definition 13. A Unique Factorization Domain is an integral domain R in which every nonzero nonunit element r 2 R satisfies the following: (1) r can be written as a finite product of irreducibles r = p1p2 ··· pn, and (2) the decomposition is unique in the following sense: if r = q1q2 : : : qm is another factorization of r into irreducibles, then m = n and we can reorder the factors so that for each i, pi = uiqi for a unit ui. Example. (1) Z, R[x] ifpR is a UFD. p (2) Nonexample:p Z[ −5]. We have two factoriaztions 6 = 2 · 3 = (1 + −5)(1 − −5)..