MODERN ALGEBRA (MATH 210) PROBLEM SET 3

1. (a) Must any finite integral domain be a field? (b) (The field 𝔽_{4}.) Ex. 3.14 (p. 125). Ex. 3.44 (ii) (p. 149).

2. Prove that the quotient ring ℤ[i]/I is finite for any nonzero ideal I of ℤ[i]. (Hint: Use the fact that I = (α) for some nonzero α and then use the division algorithm to see that every coset of I is represented by an element of norm less than N(α).)

3.

(a) Prove that if an integer is the sum of two rational squares, then it is the sum of two integer squares (for example, 13 = (1/5)^{2} + (18/5)^{2} = 2^{2} + 3^{2}).

(b) Determine all the representations of the integer 2130797 = 17^{2} · 73 · 101 as the sum of two squares.

(c) Find a generator for the ideal (47 − 13i, 53 + 56i) in ℤ[i].

4. Let I and J be ideals of a commutative ring R.

(a) Prove that I + J is the smallest ideal of R containing both I and J.

(b) Prove that IJ is an ideal contained in I ∩ J.

(c) Given an example where IJ ≠ I ∩ J.

(d) Prove that if I + J = R then IJ = I ∩ J.

5. A commutative ring R is called a local ring if it has a unique maximal ideal. Prove that if R is a local ring with maximal ideal 𝔪 then every element of R − 𝔪 is a unit. Prove conversely that if R is a commutative ring with 1 in which the set of nonunits forms an ideal 𝔪, then R is a local ring with unique maximal ideal 𝔪.

6. Let ω be the cube-root of unity \frac{−1+\sqrt{−3}}{2}. Show that the ring R = ℤ[ω] (sometimes called the Eisenstein integers) has unique factorization. (Hint: see the proof for the Gaussian integers in Sec. 3.6.) Factor 2, 3, 5, and 7 into primes in R. (Which one of them has a repeated prime factor? This prime factor is key to an elementary proof of Fermat’s Last Theorem for n = 3. If you’d like to see it, just ask me.)

7. Let K be a field. A discrete valuation on K is a function v : K^{×} → ℤ satisfying

(i) v(ab) = v(a) + v(b) (i.e., v is a homomorphism from the multiplicative group of nonzero elements of K to ℤ),

(ii) v is surjective, and (iii) v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K^{×} with x + y ≠ 0.

Date: Tuesday, October 22, 2002.

The set R = {x × K^{×} : v(x) ≥ 0} ∪ {0} is called the valuation ring of v.

(a) Prove that R is a subring of K which contains the identity. (In general, a ring R is called a discrete valuation ring if there is some field K and some discrete valuation v on K such that R is the valuation ring of v. In fact K = Frac(R).)

(b) Prove that for each nonzero element x ∈ K either x or x^{−1} is in R.

(c) Prove that an element x is a unit of R if and only if v(x) = 0.

(d) The p-adic valuation v_{p}. Let p be a prime. One example of a discrete valuation is v_{p} : ℚ^{×} → ℤ given by v_{p}(a/b) = α where a/b = p^{α}c/d, with (p, cd) = 1. Describe the element of the corresponding valuation ring. Describe the units of the valuation ring.

Remark: There is a theorem that the only valuations on ℤ are these and the usual one. You can do analysis on ℚ using these valuations; they and their generalizations are central in number theory.

8. Let F be a field. Define the ring F ((x)) of formal Laurent series with coefficients in F by

F ((x)) = {∑_{n≥N}^{∞} a_{n}x^{n} : a_{n} ∈ F, N ∈ ℤ} .

(Every element of F ((x)) is a power series in x plus a polynomial in 1/x, i.e. each element of F ((x)) has only a finite number of terms with negative powers of x.)

(a) Prove that F ((x)) is a field.

(b) Define a discrete valuation on F ((x)) whose discrete valuation ring is F [[x]], the ring of formal power series with coefficients in F.

Hence F ((x)) = Frac(F [[x]]). (This can be shown directly as well.)

9. Let D be a squarefree integer, and let O = ℤ[ω] be the ring of integers in the quadratic field ℚ(\sqrt{D}), where ω = (1 + \sqrt{D})/2 if D ≡ 1 (mod 4), and ω = \sqrt{D} otherwise. For any positive integer f prove that the set O_{f} = ℤ[fω] = {a + bfω : a, b ∈ ℤ} is a subring of O. Prove that [O : O_{f}] = f (where the index is as additive abelian groups). Prove conversely that a subring of O containing the identity and having finite index f in O is equal to O_{f}. (The ring O_{f} is called the order of conductor f in the field ℚ(\sqrt{D}). The ring of integers O is called the maximal order in ℚ(\sqrt{D}).)

10. An example of a non-commutative ring. Suppose G is a finite group. The group ring of G with coefficients in a commutative ring R, denoted RG, is defined on p. 521 (where it is called a group algebra). Warning: this is a non-commutative ring if G is not abelian!

1. Let K = {k_{1}, …, k_{m}} be a conjugacy class in G. Prove that the element K := k_{1} + ⋯ + k_{m} is in the center of RG.

2. Let K_{1}, …, K_{r} be the conjugacy classes of G, and for each K_{i} let K_{i} be the element of RG that is the sum of the members of K_{i}. Prove that an element α ∈ RG is in the center of RG if and only if α = a_{1}K_{1} + a_{2}K_{2} + ⋯ + a_{r}K_{r} for some a_{1}, …, a_{r} ∈ R.

11. Let R be the ring of all continuous functions from ℝ to ℝ, and for each c ∈ ℝ let M_{c} be the ideal {f ∈ R : f(c) = 0}.

(a) Show that M_{c} is a maximal ideal.

(b) Let I be the collection of functions f(x) in R with compact support (i.e. f(x) = 0 for |x| sufficiently large) Prove that I is an ideal of R that is not a prime ideal.

(c) Let M be a maximal ideal of R containing I. Prove that M ≠ M_{c} for any c ∈ ℝ.

12. A universal side divisor u is a non-zero, non-unit element of R such that for every x ∈ R, either u|x or u|(x + z) for some unit z.

A Euclidean domain has universal side divisors: fix the norm N, and let u be an element of R − {0, units} of minimal norm. Then for any x ∈ R, write x = qu + r where r = 0 or N(r) < N(u) (i.e. r is a unit). Thus u divides x in the first case, and x − r in the second. (This argument is given in Prop. 3.63 on p. 154.)

Suppose R = ℤ[α] where α = (1 + \sqrt{−19})/2. Prove that R is not a Euclidean domain by showing that it has no universal side divisors as follows.

(a) Find the units of R.

(b) Take the usual norm N on R (N(z) = z\overline{z}). Show that the only elements of norm less than 5 are {0, ±1, ±2}.

(c) Taking x = 2 in the definition of a universal side divisor, show that any universal side divisor must be a nonunit divisor of 2 of 3.

(d) Find all nonunit divisors of 2 and 3.

(e) Now taking x = α in the definition of universal side divisor, find a contradiction.

(f) Repeat the argument for α = (−1 + \sqrt{−163})/2.

It is not hard to show that ℚ(α) is a Principal Ideal Domain, so this shows that not every Principal Ideal Domain is a Euclidean Domain.

The set is due Tuesday, October 29 at 3:30 pm in Pierre Albin’s mailbox (opposite the elevator on the first floor of Building 380).