Math 6310, Fall 2017 Homework 10 1. Let R be an integral domain and p 2 R n f0g. Recall that: (i) p is a prime element if and only if (p) is a prime ideal. (ii) p is an irreducible element if and only if (p) is maximal among the proper principal ideals. Let R be a PID. (i) Show that R is a UFD. (ii) Show that every proper nonzero prime ideal in R is maximal. 2. Show that the polynomial ring Z[x] is not a PID. On the other hand, Z[x] is a UFD, as we will see in class shortly. 3. Let R be an integral domain and a; b 2 R n f0g. An element d 2 R n f0g is called a greatest common divisor of a and b if the following conditions hold: (i) d j a and d j b; (ii) if e 2 R n f0g is another element such that e j a and e j b, then also e j d. In this case we write d 2 gcd(a; b). (a) Show that d 2 gcd(a; b) () (d) is the smallest principal ideal containing (a; b). (b) Suppose d 2 gcd(a; b). Show that d0 2 gcd(a; b) () d ∼ d0. (c)( Bezout's theorem) Suppose R is a PID. Show that there exist r; s 2 R such that ra + sb 2 gcd(a; b): In an arbitrary integral domain, greatest common divisors may not exist. When they do, they are unique up to associates, by part (b). 4. Let R be a UFD and a; b 2 R n f0g. (a) Explain why we can always write α1 αk β1 βk a = up1 ··· pk and b = vp1 ··· pk where u and v are units, k ≥ 0, fp1; : : : ; pkg are irreducibles no two of which are associate, and αi; βi ≥ 0. (b) Given a and b as in (a), show that a j b () αi ≤ βi 8 i: (c) Given a and b as in (a), let γ1 γk d := p1 ··· pk where γi = minfαi; βig. Show that d 2 gcd(a; b). Thus greatest common divisors always exist in a UFD. 1 (d) Let c 2 R n f0g. Suppose that a j bc and 1 2 gcd(a; b). Show that a j c. (e) Let d 2 gcd(a; b) and c 2 R n f0g. Show that cd 2 gcd(ca; cb). (f) Let d 2 gcd(a; b). Write a = da0 and b = db0. Show that 1 2 gcd(a0; b0). 5. Let R be a UFD with field of fractions F . (a)( Rational Root Theorem) Let n f(x) = a0 + a1x + ··· + anx be a polynomial in R[x] of degree n. Let α 2 F be a root of f(x), write α = a=b with 1 2 gcd(a; b). Show that aja0 and bjan. (b) Deduce that R is integrally closed. 6. Let R be a UFD with field of fractions F . Prove that the quotient group F ×=R× is a free abelian group. [You need to show that every element of F ×=R× is uniquely a product (with integer exponents) of elements from an appropriate set of generators.] 7. Our study of unique factorization made heavy use of the following fact about integral domains: (a) = (b) if and only if there exists a unit u such that a = ub. Give an example to show that this can fail in general commutative rings. 8. Let R be a subring of R, the field of real numbers. Let a 2 R and b 2 R n f0g. Consider the following argument. −1 1 • There exists q 2 Z such that jb a − qj ≤ 2 . • Hence a = bq + r for some r 2 R with jrj < jbj. • Hence R is a Euclidean domain with norm δ(x) = jxj. p • In particular, Z[ d] is a Euclidean domain for any positive integer d. We know the conclusion is false. What is wrong with the argument? p p p 9. Let d 2 Z be an integer such that d2 = Z. Let Q( d) := fa + b d 2 C : a; b 2 Qg. p (a) Show that ( d) is a subfield of . Q p C (b) Let a; b 2 . Show that a + b d = 0 () a = b = 0. Q p p (c) The Galois conjugate of α = a + b d is α := a − b d. p p b (i) Show that the map Q( d) ! Q( d), α 7! αb, is well-defined. (ii) Show that it is a field automorphism. p p 2 2 (d) The field norm is the map N : Q( d) ! Q given by N(a + b d) := a − b d. (i) Show that N(α) = ααb. (ii) Show that N(α) = 0 () α = 0. (iii) Show that N(αβ) = N(α)N(β). (iv) Suppose d < 0. Show that α = α and N(α) = jαj2. p p b p p (e) Let Z[ d] = fa + b d 2 C : a; b 2 Zpg. Note that Z[ d] is a subring of Q( d) (hence an integral domain). Let α 2 [ d]. Z p (i) Show that N(α) = ±1 () α is a unit in in [ d]. Z p (ii) Show that if N(α) is prime in Z then α is irreducible in Z[ d]. 2 (f) Let D be a square-free integer. Let (p D if D ≡ 2; 3 mod 4; p ! := −1+ D 2 if D ≡ 1 mod 4; and O(D) := Z[!] = fa + bw 2 C : a; b 2 Zg. p (i) Show thatp O(D) is a subring of Q( D) (hence an integral domain) and note that [ D] ⊆ O(D). Z p (ii) Let α = a + bw 2 O(D). Show that the field norm of Q( D) satisfies ( a2 − b2D if D ≡ 2; 3 mod 4; N(α) = 2 2 1−D a − ab + b ( 4 ) if D ≡ 1 mod 4: (iii) Deduce that N(α) 2 Z and, if D < 0, N(α) 2 N. (iv) Note that the assertions of 9e continue to hold for O(D). 10. Consider the integral domain O(−1) = Z[i] of Gaussian integers and the field norm N of Exercise9. (a) Show that the group of units of Z[i] is {±1; ±ig, the group of 4-th roots of unity (the Klein group). (b) Show that 1 + i and 1 + 2i are irreducible in Z[i]. (c) Show that 2 and 5 are not irreducible in Z[i]. (d) Show that 3 is irreducible in Z[i] even though N(3) is not prime. Gaussian primes (from Mathworld and Wikimedia). 3 p 11. Let d be a negative integer. Consider the integral domain Z[ d] and the field norm N of Exercise9. p 2 1+jdj (a) Given z 2 C, show that there exists γ 2 Z[ d] such that jz − γj ≤ . p 4 (b) Show that if dp= −1 or d = −2 then N : Z[ d] ! N is a Euclidean norm and deduce that Z[ d] is a Euclidean domain. p 12. Consider the integral domain Z[ −3] and the field norm N of Exercise9. p (a) Show that the units of Z[ −3] are 1 and −1. p (b) Show that there is no β 2 Z[ −3] such that N(β) = 2. p (c) Let α 2 Z[ −3]. Show that if N(α) = 4 then α is irreducible. p p (d) Show that 2 and 1 + −3 are irreducible in Z[ −3]. p (e) Show that 4 2 Z[ −3] admits two distinct decompositions into irreducibles (even up to units and reordering). p p (f) Deduce (again) that Z[ −3] is not a UFD and that N : Z[ −3] ! N is not a Euclidean norm. p −1+i 3 13. Consider the integral domain O(−3) = Z[!] of Eisenstein integers, where ! = 2 , a primitive cube root of unity. (a) By Exercise 9(f)iii, the field norm restricts to N : O(−3) ! N. Show this is a Euclidean norm on O(−3). 2 (b) Show that the group of units of Z[!] is {±1; ±!; ±! g, the group of 6-th roots of unity. (c) Show that 2 + ! and 2 − ! are irreducible in Z[!]. (d) Show that 3 and 7 are not irreducible in Z[!]. (e) Show that 2 is irreducible in Z[!] even though N(2) is not prime. Eisenstein primes (from Mathworld and Wikipedia). 4 p 14. Consider the integral domain O(−5) = Z[ −5]. (a) Find the group of units. (b) Find α 2 O(−5) n Z such that 9 = αα (c) Let β 2 O(−5). Show that if N(β) = 9 then β is irreducible. (d) Deduce that 3, α and α are irreducible are no two of them are associate. (e) Deduce that neither 3, α nor α is prime, and that O(−5) is not a UFD. 15. This is a continuation of 14. Let P = (3; α), P = (3; α). (a) Find a surjective homomorphism of rings f : O(−5) ! F3 such that Ker(f) = P . (b) Deduce that P and P are maximal (hence prime) ideals of O(−5). 2 (c) Show that (3) = P P ,(α) = P 2,(α) = P . (d) Observe that the two distinct irreducible factorizations 9 = 32 = αα lead to the same prime ideal factorization 2 (9) = (P P )2 = P 2P : For any square-free D, the ring O(D) is a Dedekind domain. In these domains, any ideal admits a factorization into prime ideals, unique up to order of the factors.