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Math 6310, Fall 2017 Homework 10

1. Let R be an integral and p ∈ R \{0}. Recall that:

(i) p is a prime element if and only if (p) is a prime . (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 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 and a, b ∈ R \{0}. An element d ∈ R \{0} is called a of a and b if the following conditions hold:

(i) d | a and d | b; (ii) if e ∈ R \{0} is another element such that e | a and e | b, then also e | d.

In this case we write d ∈ gcd(a, b).

(a) Show that d ∈ gcd(a, b) ⇐⇒ (d) is the smallest containing (a, b). (b) Suppose d ∈ gcd(a, b). Show that d0 ∈ gcd(a, b) ⇐⇒ d ∼ d0. (c)( Bezout’s theorem) Suppose R is a PID. Show that there exist r, s ∈ R such that

ra + sb ∈ 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 ∈ R \{0}.

(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, {p1, . . . , pk} are irreducibles no two of which are associate, and αi, βi ≥ 0. (b) Given a and b as in (a), show that

a | b ⇐⇒ αi ≤ βi ∀ i.

(c) Given a and b as in (a), let γ1 γk d := p1 ··· pk

where γi = min{αi, βi}. Show that d ∈ gcd(a, b). Thus greatest common divisors always exist in a UFD.

1 (d) Let c ∈ R \{0}. Suppose that a | bc and 1 ∈ gcd(a, b). Show that a | c. (e) Let d ∈ gcd(a, b) and c ∈ R \{0}. Show that cd ∈ gcd(ca, cb). (f) Let d ∈ gcd(a, b). Write a = da0 and b = db0. Show that 1 ∈ 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 ∈ F be a root of f(x), write α = a/b with 1 ∈ gcd(a, b). Show that a|a0 and b|an. (b) Deduce that R is integrally closed.

6. Let R be a UFD with field of fractions F . Prove that the quotient F ×/R× is a free . [You need to show that every element of F ×/R× is uniquely a product (with exponents) of elements from an appropriate set of generators.]

7. Our study of unique made heavy use of the following fact about integral : (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 of R, the field of real numbers. Let a ∈ R and b ∈ R \{0}. Consider the following argument.

−1 1 • There exists q ∈ Z such that |b a − q| ≤ 2 . • Hence a = bq + r for some r ∈ R with |r| < |b|. • Hence R is a Euclidean domain with norm δ(x) = |x|. √ • 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? √ √ √ 9. Let d ∈ Z be an integer such that d∈ / Z. Let Q( d) := {a + b d ∈ C : a, b ∈ Q}. √ (a) Show that ( d) is a subfield of . Q √ C (b) Let a, b ∈ . Show that a + b d = 0 ⇐⇒ a = b = 0. Q √ √ (c) The Galois conjugate of α = a + b d is α := a − b d. √ √ b (i) Show that the map Q( d) → Q( d), α 7→ αb, is well-defined. (ii) Show that it is a field automorphism. √ √ 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) = |α|2. √ √ b √ √ (e) Let Z[ d] = {a + b d ∈ C : a, b ∈ Z√}. Note that Z[ d] is a subring of Q( d) (hence an integral domain). Let α ∈ [ d]. Z √ (i) Show that N(α) = ±1 ⇐⇒ α is a unit in in [ d]. Z √ (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 (√ D if D ≡ 2, 3 mod 4, √ ω := −1+ D 2 if D ≡ 1 mod 4,

and O(D) := Z[ω] = {a + bw ∈ C : a, b ∈ Z}. √ (i) Show that√ O(D) is a subring of Q( D) (hence an integral domain) and note that [ D] ⊆ O(D). Z √ (ii) Let α = a + bw ∈ 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) ∈ Z and, if D < 0, N(α) ∈ 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 and the field norm N of Exercise9.

(a) Show that the group of units of Z[i] is {±1, ±i}, 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 √ 11. Let d be a negative integer. Consider the integral domain Z[ d] and the field norm N of Exercise9. √ 2 1+|d| (a) Given z ∈ C, show that there exists γ ∈ Z[ d] such that |z − γ| ≤ . √ 4 (b) Show that if d√= −1 or d = −2 then N : Z[ d] → N is a Euclidean norm and deduce that Z[ d] is a Euclidean domain. √ 12. Consider the integral domain Z[ −3] and the field norm N of Exercise9. √ (a) Show that the units of Z[ −3] are 1 and −1. √ (b) Show that there is no β ∈ Z[ −3] such that N(β) = 2. √ (c) Let α ∈ Z[ −3]. Show that if N(α) = 4 then α is irreducible. √ √ (d) Show that 2 and 1 + −3 are irreducible in Z[ −3]. √ (e) Show that 4 ∈ Z[ −3] admits two distinct decompositions into irreducibles (even up to units and reordering). √ √ (f) Deduce (again) that Z[ −3] is not a UFD and that N : Z[ −3] → N is not a Euclidean norm. √ −1+i 3 13. Consider the integral domain O(−3) = Z[ω] of Eisenstein integers, where ω = 2 , a primitive cube .

(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[ω] is {±1, ±ω, ±ω }, the group of 6-th roots of unity. (c) Show that 2 + ω and 2 − ω are irreducible in Z[ω]. (d) Show that 3 and 7 are not irreducible in Z[ω]. (e) Show that 2 is irreducible in Z[ω] even though N(2) is not prime.

Eisenstein primes (from Mathworld and Wikipedia).

4 √ 14. Consider the integral domain O(−5) = Z[ −5]. (a) Find the group of units. (b) Find α ∈ O(−5) \ Z such that 9 = αα (c) Let β ∈ 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

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 . In these domains, any ideal admits a factorization into prime ideals, unique up to order of the factors. The notion of ideal was proposed in the work of Dedekind around 1876. It was a generalization of the concept of ideal numbers of Kummer. They realized that replacing elements by ideals, uniqueness of prime factorizations is recovered. This finding constitutes the starting point of . A Dedekind domain is an integral domain that is noetherian and integrally closed, and in which every prime ideal is maximal. Every PID is a Dedekind domain. A UFD is a Dedekind domain if and only if it is a PID.

16. Let R = {p(x) ∈ Q[x] | p(0) ∈ Z}. Note that R is a subring of Q[x], hence an integral domain. Show that R does not satisfy the ascending chain condition on principal ideals. In particular, R is not a UFD. Recall that a domain is a UFD if and only if it satisfies ACC and all irreducibles are primes. It is a fact that every irreducible element of R is prime. So this condition alone is not sufficient to have a UFD.

17. Let R be an integral domain. Show the following statements are equivalent.

(i) R is a field. (ii) Any function δ : R \{0} → N is a Euclidean norm on R. (iii) Any constant function δ : R \{0} → N is a Euclidean norm on R.

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