Reminder: The "How to" guide for proof writing, including proper referencing of prior results, should be followed when writing up your solutions.
Unless otherwise specified, the problem set is due at the start of class on the due date.

• Problem Set 1:
  • E5.10: Let C₃ = ⟨ a | a³ = 1 ⟩ and let R be the group ring (Z/2Z)C₃.
    (a) Let S = (Z/2Z)[y]/(y³-1) and T = Z[x]/(2,x³-1).
    (a-i) Show that R and S are isomorphic rings.
    (a-ii) show that S and T are isomorphic rings.
    (b) Write out the elements of R, and determine which of them are units and which are zero divisors.
    (c) Let I be the augmentation ideal of R. Find all of the elements of I, show that I is a principal ideal, and determine whether the ideal I is prime or maximal (or neither).
    (d) Repeat (c) for the ideal J = (1+a+a²).
  • E5.11: Let R be an integral domain, let r be a nonzero element of R, and let I = (r). Show that if I is a prime ideal, then r is an irreducible element of the ring R.
    (Recall from E5.6 that an element r of an integral domain is defined to be irreducible if r is not a unit, and whenever r = xy for some x,y ∈ R, then either x or y is a unit.)
  • E5.12: Let R be a commutative ring with 1 ≠ 0. Prove that the set of prime ideals of R has a minimal element with respect to inclusion.
    (Hint: Use Zorn's Lemma/Axiom 5.141.)
  • E5.13: Let R be a commutative ring with 1 ≠ 0. The Binomial Theorem says that (r+s)ⁿ = ∑_i=0ⁿ b(n,i)rⁱs^n-i for all r,s ∈ R and n ≥ 0, where b(n,i) is the sum 1 + 1 + ··· + 1 of (n!)/(i!)((n-i)!) copies of 1 in R. (You don't need to prove this.) Let I be an ideal of R.
    (a) The radical of I is the set rad(I) = {r ∈ R | rⁿ ∈ I for some n ≥ 1}. Use the Binomial Theorem to show that rad(I) is an ideal of R.
    (b) The ideal I is a radical ideal if rad(I) = I. Show that every prime ideal of R is a radical ideal.
    (c)The nilradical of R is the ideal rad(0); that is, set of nilpotent elements of R, denoted N(R).
    (c-i) Show that if R/N(R) is a field, then every element of R is either nilpotent or a unit.
    (c-ii) Show that if every element of R is either nilpotent or a unit, then R has exactly one prime ideal, and that prime ideal is N(R).
  • E5.14: (a) Prove the HBT for rings of fractions (TOC Thm 5.152).
    (b) Let R be an integral domain and let S be a nonempty subset of R that is closed under multiplication such that 1 ∈ S and 0 ∉ S. Prove that the ring of fractions S^-1R is isomorphic to a subring of the field of fractions of R (and hence S^-1R is also an integral domain).
  • E5.15: Let R be a commutative ring with 1 ≠ 0, and let r ∈ R be a nonzerodivisor. Let R[1/r] denote the ring S^-1R where S = {rⁿ | n ≥ 0}. Prove that R[1/r] ≅ R[x]/(rx-1).
  Due 1/19/18 for grading: E5.10(a-i,b,c,d), E5.11, E5.13(c), E5.14(b)

• Problem Set 2:
  • E5.16: (a) Let R be a commutative ring with 1 ≠ 0, let I be an ideal of R, and let (I) be the ideal of R[x] generated by I. Then R[x]/(I) ≅ (R/I)[x]. (That is, prove TOC Thm 5.93.)
    (b) For any polynomial f(x) ∈ Z[x] and natural number n, let f(x) mod n denote the polynomial in (Z/nZ)[x] obtained by replacing the coefficients in the polynomial f(x) by their (standard) images in Z/nZ. Show that if m and n are relatively prime natural numbers and f(x) and g(x) are polynomials in Z[x] of degree d, then there is a polynomial h(x) in Z[x] of degree d such that h(x) mod m = f(x) mod m and h(x) mod n = g(x) mod n.
  • E6.1: Let R be a commutative ring with 1 ≠ 0 and let a,b ∈ R with b ≠ 0. A least common multiple, or lcm, of a and b is an element m ∈ R satisfying a | m, b | m, and whenever [a | m' and b | m'] then m | m'. Show that if a and b have a lcm m, then the ideal (m) is the unique largest principal ideal contained in (a) ∩ (b).
  • E6.2: (a) Is every subdomain (that is, subring that is an integral domain) of a Euclidean domain a Euclidean domain?
    (b) Is every product of Euclidean domains a Euclidean domain?
    (c) Is every quotient of a Euclidean domain also a Euclidean domain?
  • E6.3: Let R be the quadratic integer ring Z[√(-2)] (a subring of C).
    (a) Let S = Z[x]/(x²+2). Show that R ≅ S.
    (b) Let T be the subring of M₂(Z) consisting of the matrices with top row [a b] and bottom [-2b a] for all a,b ∈ Z. Show that R ≅ T.
    (c) Show that R is an integral domain.
    (d-i) Recall that the restriction to R of the complex norm, N: R → Z (where N(a +b√(-2)) = a² + 2b²) satisfies: [N(r) = 0 if and only if r = 0] and [N(rs) = N(r)N(s) for all r,s ∈ R]. Using the complex norm, show that R is a Euclidean domain. (Hint: Mimic the proof that Z[i] is a Euclidean domain in the text in section 8.1.)
    (d-ii) Use the division algorithm in the Euclidean domain Z[i] to find a generator for the ideal (85, 1+13i) in the ring Z[i].
  • E6.4: Let R be a commutative ring with 1 ≠ 0. Let < be the strict partial order on the set A of ideals of R defined by I < J if and only if I ⊋ J. The ring R is called Noetherian, or satisfies the ascending chain condition (acc), if there is no infinite ascending chain of ideals I₁ ⊊ I₂ ⊊ I₃ ⊊ ...; that is, if the strict partial order is well-founded. Show that the ring R is Noetherian if and only if every ideal of R is finitely generated.
  Due 1/26/18 for grading: E5.16(b), E6.1, E6.3(c,d-ii), E6.4

• Problem Set 3:
  • E6.5: (a) Give, and prove, an example of a PID R for which R[x] is not a PID.
    (b-i) Show that if T and T' are commutative rings with 1 ≠ 0 then every ideal of T × T' is of the form I × I' where I is an ideal of T and I' is an ideal of T'.
    (b-ii) Show that if R and S are PID's, then all ideals in R × S are principal.
    (c) Show that if R is a PID and P is a prime ideal of R, then R/P is a PID.
  • E6.6: Let F be a field, and let p ∈ F[x].
    (a) Prove that F[x]/(p) is a field if and only if p is irreducible.
    (b) Let a ∈ F. The element a of F is called a root of p if p(a) = 0. Show that the remainder in the division of p by x-a is p(a), and consequently x-a divides p if and only if a is a root of p.
    (c) Suppose that the degree of the polynomial p is n ≥ 1.
    (c-i) Show that for all q ∈ F[x] there is a q' ∈ F[x] such that degree(q') ≤ n-1 and q + (p) = q' + (p).
    (c-ii) Show that if q,r ∈ F[x] are polynomials of degree ≤ n-1 and q + (p) = r + (p), then q = r.
  • E6.7: List (and prove) all of the ideals of the ring Z[x] that contain the ideal (2,x³+1).
  • E6.8: Let R be a UFD and let a,b ∈ R. Show that R contains a gcd and an lcm of a and b.
  • E6.9: Let R = Z[√(-5)].
    (a) Let p be a prime element in Z with p > 0. Show that p is a prime element in R if and only if the polynomial x²+5 in Z/pZ[x] does not have a root in Z/pZ.
    (b) For each of the integers 7 and 13, determine whether number is prime in R.
  • E6.10: Let R = Z[√(-n)] where n is a squarefree integer greater than 3.
    (a) Prove that √(-n) and 1+√(-n) are irreducibles in R.
    (b) Prove that R is not a UFD. [Hint: Show that either √(-n) or 1+√(-n) is not a prime element.]
  Due 2/2/18 by 3:30pm for grading: E6.6(a), E6.7, E6.9(a), E6.10

• Problem Set 4:
  • E6.11: Let F be a field and let p ∈ F[x] be a polynomial with positive degree. Show that there exists a field E and an embedding i: F → E such that if j: F[x] → E[x] is the embedding induced (using the HBT for polynomial rings) by the homomorphism i and the map {x} → E[x] sending x to x, then the polynomial j(p) in E[x] has a root in E[x].
  • E6.12: For each of the following rings R and polynomials p, determine (and prove) whether the polynomial p is irreducible in R.
    (a) R = Q[x,y] and p = x² + y² - 1.
    (b) R = Q[x] and p = x⁴ + x + 1.
    (c) R = Z/2Z[x] and p = x⁵ + x + 1. (Hint: First find all of the irreducible polynomials in R of degree 2 or 3 - it's a very short list!)
    (d) R = (Z[i])[x] = (Z[√(-1)])[x] and p = x² + i.
  • E6.13: (a) Show that (Z/2Z)[x]/(x²+x+1) is a field with 4 elements.
    (b) Construct fields of orders 8, 9, and 49, respectively. (Prove your answer.)
    (c) Find a generator of the multiplicative group of nonzero elements in your field of order 9 from part (b).
    (d) Show that for every prime number p there is a field of order p².
  • E6.14: Let R be a commutative ring with 1 ≠ 0. For any polynomial p ∈ R[x], the content of p, denoted cont(p), is the ideal in R generated by the coefficients of p. A polynomial p ∈ R[x] is called primitive if cont(p) = R. Let f,g ∈ R[x].
    (a) Prove that cont(fg) ⊆ cont(f)cont(g).
    (b) Let P be a prime ideal in R. Show that cont(fg) ⊆ P if and only if cont(f)cont(g) ⊆ P.
    (c) Show that f and g are both primitive if and only if fg is primitive.
  • E6.15: Let R = Z/6Z[x] and let f ∈ R be the polynomial f(x) = x.
    (a) Show that f = pq where p = 3x + 4 and q = 4x + 3 are in R, and hence f is reducible.
    (b) Let I and J be the ideals I = (2) and J = (3) in R. Show that f + I is irreducible in R/I, and f + J is irreducible in J.
    (c) Explain why parts (a) and (b) do not contradict TOC Prop 6.84.
  • E6.16: Let R = Q[x,y]. Order the monic monomials in R by the degree-lex order: x^ay^b > x^cy^d if and only if either [a+b > c+d] or [a+b = c+d and a < c].
    (a) Find Groebner bases for the following ideals I of R:
    (a-i) I = (x² + y² + 1, x - y)
    (a-ii) J = (x³ - y, x²y - x)
    (b) Use your Groebner basis for the ideal J in (a-ii) to determine whether or not each of the following polynomials is in J: p = x²y - y³, q = x⁵y² - x⁴y + x³ - y²
    (c) Show that the Groebner basis for the ideal K = (x²+x+1, x+1) contains the rewriting rule 1 → 0. Use your proof to find r,s ∈ R such that 1 = r(x² + x + 1) + s(x + 1).
  Due 2/9/18 for grading: E6.11, E6.12(a,b), E6.13(d), E6.16(a-ii,b)

• Problem Set 5:
  • E7.1/E6.17: Let R be a commutative ring with 1 ≠ 0. For any R-modules M,N in this problem, let Hom_R(M,N) have the R-module structure in TOC Prop 7.12. Also all instances of ≅ in this problem denote R-module isomorphisms. Let A,B,M,N be R-modules.
    (a) Prove that Hom_R(R,N) ≅ N.
    (b) Prove that Hom_R(A × B,N) ≅ Hom_R(A,N) × Hom_R(B,N), and hence Hom_R(Rⁿ,N) ≅ Π_i=1ⁿ N.
    (c) Let I be an ideal of R, and let N' = {n ∈ N | in = 0 for all i ∈ I}. Show that N' is an R-submodule of N, and that Hom_R(R/I,N) ≅ N'.
    (d) Show that if m,n are natural numbers and R = Z, then Hom_Z(Z/mZ,Z/nZ) ≅ Z/dZ where d = gcd(m,n).
  • E7.2/E6.18: Let R be a ring with 1 ≠ 0 and let M be an R-module.
    (a) Show that the annihilator ann_R(N) = {r ∈ R | rn = 0 for all n ∈ N} is a (2-sided) ideal of R.
    (b) Find the natural number n satisfying ann_Z(Z/12Z × Z/10Z) = (n).
  • E7.3/E6.19: Let F be a field, let V,W be vector spaces over F, and let a: V → V and b:W → W be linear transformations. Let V_a denote the F[x]-module that is the vector space V with the unique F[x]-action satisfying xv = a(v) for all v ∈ V. [That is, (r_nxⁿ + ··· + r₀)v = r_naⁿ(v) + ··· + r₀v for all r_nxⁿ + ··· + r₀ ∈ F[x].) Similarly let W_b denote the F[x]-module that is the vector space W with the unique F[x]-action satisfying xw = b(w) for all w ∈ W.
    (a) Show that a function g: V_a → W_b is an F[x]-module homomorphism if and only if (1) g: V → W is a linear transformation and (2) g ∘ a = b ∘ g.
    (b) Suppose that V = F^m = W, and let A,B ∈ M_m,m(F) be the matrices representing the linear transformations a and b, respectively. Show that there is an F[x]-module isormorphism V_a ≅ W_b if and only if the matrices A and B are similar matrices: that is, there is an invertible matrix P such that B = PAP^-1.
  • E7.4/E6.20: An R-module N is irreducible if N ≠ 0 and the only submodules of N are 0 and N. An R-module N has length k if there is a chain of submodules 0 = N₀ ⊂ N₁ ⊂ ··· ⊂ N_k = N such that each quotient module N_i+1/N_i is irreducible. If there is no such finite chain of submodules, the length of N is defined to be infinity. (The length of an R-module is well-defined.) Let R be a commutative ring with 1 ≠ 0 and let M,N be R-modules.
    (a) Show that M is irreducible if and only if there is a maximal ideal I of R such that M ≅ R/I (isomorphic as R-modules).
    (b) Show that length(M ⊕ N) = length(M) + length(N).
  • E7.5/E6.21: Let R be a ring with 1 ≠ 0, let M be an R-module, and let N be an R-submodule of M.
    (a) Show that if M is finitely generated as an R-module, then so is M/N.
    (b) Show that if N and M/N are finitely generated as R-modules, then so is M.
  • EE7.6/6.22: Let R be a commutative ring with 1 ≠ 0. For any ideal J of R and R-module N, define JN = {∑_k=1ⁿ j_kn_k | n ≥ 0, j ∈ J, n ∈ N}. Let I be an ideal of R, and let M be an R-module.
    (a) Show that IM is an R-submodule of M and show that M/IM is an (R/I)-module with ring action (r+I)(m+IM) = rm+IM for all r+I ∈ R/I and m+IM ∈ M/IM.
    (b) Show that if I is any ideal of R, then Rⁿ/IRⁿ ≅ (R/I)ⁿ as (R/I)-modules.
    (c) For any field F and natural numbers m,n, a theorem that we will prove in class soon (and that you can use in this problem now) says that F^m ≅ Fⁿ (as F-vector spaces) if and only if m = n. Show that R^m ≅ Rⁿ (as R-modules) if and only if m = n.
  Due 2/16/18 for grading: E6.17(a,b), 6.19(a), 6.20(a), 6.22(b,c)

• Problem Set 6:
  • E7.7/E6.23: (a) Let R be any ring with 1 ≠ 0. Show that if M_α is a free R-module for all α in an index set J, then the direct sum ⊕_{α ∈ J} M_α is a free R-module.
    (b) Show that if V is a vector space over a field F with basis B, then V ≅ ⊕_{b ∈ B} F. [Note: You may not assume that B is finite.]
  • E7.8/E6.24: Let V be a vector space over a field F, and let t: V → V be a linear transformation. A nonzero element v ∈ V satisfying t(v) = λ v for some λ ∈ F is an eigenvector of t with eigenvalue λ.
    (a) Let λ be an eigenvalue of t and let W = {eigenvectors of t with eigenvalue λ} ∪ {0}. Show that W is a subspace of V.
    (b) Suppose that λ₁,...,λ_k are distinct eigenvalues of t, and for each 1 ≤ i ≤ k suppose that v_i is an eigenvector of t with eigenvalue λ_i. Show that {v₁,...,v_k} is linearly independent. [Hint: Induct on k.]
  • E7.9/E6.25: Prove that if A is a divisible abelian group then A is not a free Z-module, and hence Q is not a free Z-module. [Recall from E4.22 that an abelian group A divisible if for each a ∈ A and n ∈ N, there is a a' ∈ A such that a = na'.]
  • E8.1: Let R be a ring with 1 ≠ 0, let M and N be R-submodules of an R-module L, and let P be the R-submodule of L generated by M ∪ N.
    (a) Show that P = M + N.
    (b) Show that the following are equivalent: (b-i) The function f: M ⊕ N → M + N defined by f(m,n) = m + n is an R-module isomorphism. (b-ii) M ∩ N = {0}. (b-iii) For all p ∈ P there are unique m ∈ M and n ∈ N such that p = m + n.
    (c) Let V and W be finite dimensional vector subspaces of a vector space U over a field F. Show that: (c-i) dim(V ⊕ W) = dim(V) + dim(W). (c-ii) If V ∩ W = {0} and V + W = U then U ≅ V ⊕ W.
  • E8.2: Let V be the set of continuous functions : [0,1] → R.
    (a) Show that V is a vector space over R.
    (b) A vector space W is infinite dimensional if W does not have a finite basis. Show that V is infinite dimensional. [Hint: Consider the functions 1,x,x²,...]
  Due 2/23/18 for grading: E6.23(a), E6.24(b), E8.1(b; equivalence of b-i and b-ii only), E8.2(b)

• Problem Set 7:
  • E8.3: Let F be a field. The column rank of a matrix M ∈ Mat_m,n(F) is the maximum number of linearly independent columns of M (viewed as elements of F^m). Let f: Fⁿ → F^m and let B = {b₁,...,bn} and C = {c₁,...,cm} be the standard ordered bases of Fⁿ and F^m, respectively.
    (a) Show that rank(f) equals the column rank of the matrix [f]_B^C.
    (b) Let A = [f]_B^C and let A' be a matrix in row reduced echelon form that is obtained from A (using "Gaussian elimination") by elementary row operations; that is, for each 1 ≤ i ≤ m the first nonzero entry a_iji of the i-th row of A' is 1, all other entries in the j_i-th column are 0, and j₁ < j₂ < ··· j_r, where r is the number of nonzero rows of A'. Show that a basis for the image f(Fⁿ) is given by the vectors f(b_j1),...,f(b_jr).
  • E8.4: Let F be a field and let V be a finite dimensional F-vector space.
    (a) Let f: V → V be a linear transformation satisfying f ∘ f = 0. Show that rank(f) ≤ (1/2) dim(V).
    (b) Let g: V → V be a linear transformation satisfying g ∘ g = g.
    (b-i) Show that g(V) ∩ Ker(g) = 0, and show that V is the internal direct sum V = g(V) ⊕ Ker(g). (Hint: Use E8.1.)
    (b-ii) Show that there is a basis B of V such that the matrix [g]_B^B is a diagonal matrix whose entries are all 0 or 1.
  • E8.5: Let V be a finite dimensional vector space. The determinant det(f) of a linear transformation f: V → V is defined to be det([f]_B^B) for any basis B of V.
    (a) Prove that det(f) is well-defined (that is, that det(f) is independent of the choice of basis B).
    (b) Fix a natural number n and let W be the vector space of all polynomials in R[x] of degree at most n. Let g: W → W be the linear transformation defined by g(p) = dp/dx. Compute det(g).
  • E8.6: Let F be a field and let A ∈ Mat_n,n(F). For any polynomial p = a_nxⁿ + ··· + a₀ ∈ F[x], let p(A) denote the matrix p(A) = a_nAⁿ + ··· + a₀I_n.
    The characteristic polynomial of A, denoted c_A, is the polynomial c_A = det(xI_n - A).
    A minimal polynomial of A, denoted m_A, is a monic polynomial of least degree such that m_A(A) = 0.
    (a) Show that if A is upper triangular (that is, the ij-th entry of A is 0 for all i > j), then det(A) is the product of the diagonal entries of A, and c_A(A) = 0.
    (b) Show that if c_A(A) = 0 and B is similar to A, then c_B(B) = 0.
    (c) Challenge problem: Prove the Cayley-Hamilton Theorem: For all A ∈ Mat_n,n(F), c_A(A) = 0. [Hint: Use E6.11 to show that F is contained in a larger field E such that c_A factors into a product of linear polynomials in E[x].]
    (Note: We'll talk about a different approach to proving the Cayley-Hamilton Theorem in class soon.)
    (d) Show that if p ∈ F[x] is a minimal polynomial of A, and q ∈ F[x] is any polynomial satisfying q(A) = 0, then p divides q. Using the Cayley-Hamilton Theorem, show that there is a unique minimal polynomial of A, and that m_A divides c_A. [Hint: Use the Euclidean algorithm.]
  • E8.7: Let R be a PID. Let A,B ∈ Mat_m,n(R), and let c and d be gcd's of the entries of A and the entries of B, respectively. Show that if B can be obtained from A by elementary row and column operations, then c and d are associates.
  • E8.8: Let R be an integral domain and let M be an R-module. The torsion submodule of M, denoted Tor(M) or Tor_R(M), is Tor(M) = {m ∈ M | rm = 0 for some r ∈ R - {0}}. Elements of Tor(M) are the torsion elements of M, and the module M is torsion-free if Tor(M) = 0.
    (a) Show that if I is a nonprincipal ideal of R, then I is a torsion-free R-module that is not a free R-module.
    (b) Show that if M and N are R-modules, then Tor(M ⊕ N) = Tor(M) ⊕ Tor(N).
    (c) Suppose that R is a PID, and that M is a finitely generated R-module. Show that if M is a torsion-free R-module then M is a free R-module.
  • E8.9: Let R = Z and let A ∈ Mat_3,4(R) be the matrix

    ⌈		1		6		5		2		⌉
    |		2		1		-1		0		|
    ⌊		3		0		3		0		⌋

    (a) Find the Smith Normal Form for A.
    (b) Let M be the R-module presented by A. Determine the invariant factor decomposition of M (that is, write M in the form in the Classification of Finitely Generated Modules over a PID).
  Due 3/9/18 for grading: E8.5(a), E8.6(a,d), E8.8(c), E8.9

• Problem Set 8:
  • E8.10: Prove TOC Lemma 8.103(2).
  • E8.11: Let F be a field.
    (a) Let n ≥ 1 and let Y,Z ∈ Mat_n,n(F). Prove that if Y and Z are similar, then Y and Z have the same minimal polynomial and the same characteristic polynomial.
    (b) Let A,B ∈ Mat_2,2(F). Prove that A and B are similar if and only if A and B have the same minimal polynomial.
    (c) Let C,D ∈ Mat_3,3(F). Prove that C and D are similar if and only if C and D have the same minimal polynomial and the same characteristic polynomial.
    (d) Give an example of two 4 × 4 matrices over F that have the same minimal polynomial and the same characteristic polynomial but are not similar.
    (e) Let p ∈ F[x] be a monic polynomial of degree n ≥ 1. Let S_p = {A ∈ Mat_n,n(F) | c_A = p} (where as usual c_A denotes the characteristic polynomial of A). Let ~ be the equivalence relation on S_p defined by A ~ B if and only if A is similar to B. Show that the set S_p is a single equivalence class if and only if the irreducible factorization of p has no repeated factors.
  • E8.12: Give a complete, non-redundant list of representatives of the conjugacy classes of GL₃(Z/2Z).
  • E8.13: (a) Show that there does not exist a matrix A ∈ Mat_3,3(Q) satisfying A⁸ = I₃ but A⁴ ≠ I₃.
    (b) Give an example of a matrix B ∈ Mat_4,4(Q) satisfying B⁸ = I₄ but B⁴ ≠ I₄.
  • E8.14: Let V = R³ with the standard basis B and let t: V → V be the linear transformation represented by the matrix [t]_B^B =

    ⌈		0		-1		0		⌉
    |		-1		0		3		|
    ⌊		0		0		1		⌋

    Find the invariant factor decomposition of the R[x]-module V_t, the characteristic and minimal polynomials of t, the eigenvalues of t, and the rational canonical form of t.
  • E8.15 Let F be a field, let V be a vector space over F with ordered basis B = {b₁,...,b_n}, let t: V → V be a linear transformation, let A = [t]_B^B, let E = {e₁,...,e_n} be the standard basis of F[x]ⁿ, let h: F[x]ⁿ → V_t be the unique F[x]-module homomorphism satisfying h(e_i) = b_i for all i, and let s: F[x]ⁿ → F[x]ⁿ be the F[x]-module homomorphism satisfying [s]_E^E = xI_n - A. Show that s(F[x]ⁿ) ⊇ Ker(h).
    (Note: This problem completes the proof of TOC Thm 8.96 from class. Hint: Consider the dimenions of the F-vector spaces F[x]ⁿ/s(F[x]ⁿ) and F[x]ⁿ/Ker(h).)
  Due 3/16/18 for grading: E8.11(d,e), E8.12, E8.13, E8.14

• Problem Set 9:
  • E8.16: Let F be a field. Let A ∈ Mat_n,n(F) and write the characteristic polynomial of A as c_A = xⁿ + b_n-1x^n-1 + ··· b₀ with each b_i ∈ F.
    (a) Show that b₀ = (-1)ⁿ det(A).
    (b) The trace of a matrix M ∈ Mat_n,n(F), denoted tr(M), is the sum of the diagonal entries of M.
    (b-i) Show that if A' is similar to A, then tr(A') = tr(A).
    (b-ii) Show that b_n-1 = -tr(A).
    (c) If the characteristic polynomial factors completely into linear factors over F, express det(A) and tr(A) in terms of the eigenvalues of A.
    (d) Show that the matrix A is diagonalizable if and only if the minimal polynomial m_A of A factors into distinct linear factors (that is, m_A = (x-λ₁) ··· (x-λ_k) for distinct elements λ₁,...,λ_k of F).
  • E8.17: For the following matrices over R, determine whether the matrix has a Jordan canonical form and if so write out the Jordan canonical form, and determine whether the matrix is diagonalizable.
    (a) A = the matrix in E8.14.

    (b) B =	⌈		2		-7		⌉
    	⌊		7		6		⌋

    (c) Let n ≥ 2 and let C_n ∈ Mat_n,n(R) be the n × n matrix in which every entry is 1. (Hint: In this part, the matrix C_n does have a Jordan canonical form.)
  • E8.18: (a) Let F be a field and let r ∈ F. Let V = Fⁿ have the standard basis E = {e₁,...,e_n}. Suppose that t: V → V is a linear transformation satisfying [t]_E^E = J_n(r). Find a basis B for V such that [t]_B^B = (J_n(r))^T (where ^T denotes the transpose of the matrix).
    (b) Let B ∈ Mat_n,n(C). Show that B is similar to the transpose B^T of B.
  • E9.1: In each part, determine the degree [Q(β) : Q]:
    (a) β = 2 + √(3)
    (b) β = 1 + ∛(2) + ∛(4).
  • E9.2: Let p be a prime number (in N), and let F = Q(i). Show that the polynomial x³ - p is irreducible over F.
  • E9.3: Let K/F be a field extension, and let a₁,...,a_m ∈ K.
    (a) Let L₀ := F and for each 1 ≤ i ≤ m let L_i := L_i-1(a_i). Show that F(a₁,...,a_m) = L_m.
    (b) For a field K with subfields K₁,...,K_m, the composite field of K₁,...,K_m, denoted K₁ ··· K_m, is the smallest subfield of K containing K₁,...,K_m. Show that F(a₁,...,a_m) = F(a₁) ··· F(a_m).
  • E9.4: Let L/F be an algebraic field extension and let a,b ∈ L.
    (a) Prove that if there is a ring automorphism h: L → L satisfying h|_F = Id_F and h(a) = b, then m_a,F = m_b,F.
    (b) Show that if m_a,F = m_b,F and L = F(a), then there is a ring automorphism h: L → L satisfying h|_F = Id_F and h(a) = b.
  Due 4/2/18 by 10:00am for grading: E8.16(d), E8.17(a,c), E9.2, E9.4

• Problem Set 10:
  • E9.5: Let K/F be a field extension of degree n. For each b ∈ K, define h_b: K → K by h_b(k) := bk for all k ∈ K.
    (a) Show that h_b is an F-linear transformation.
    (b) Show that K is isomorphic to a subring of Mat_n,n(F).
    (c) Define N_K/F : K → F by N_K/F(b) := det(h_b) for each b ∈ K.
    (c-i) Show that N_K/F induces a group homomorphism K^x → F^x and that N_K/F(a) = aⁿ for all a ∈ F.
    (c-ii) Suppose further that K = F(b) and m_b,F = xⁿ + a_n-1x^n-1 + ··· + a₀ is the minimum polynomial for b over F (with each a_i ∈ F). Show that N_K/F(b) = (-1)ⁿa₀.
  • E9.6: Let F be a field, let F' be an algebraic closure of F, and let p,q ∈ F[x]. Show that 1 is a gcd of p and q in F[x] if and only if p and q have no common roots in F'.
  • E9.7: Determine the splitting field K of the polynomial x⁶ - 4 over Q, and determine the degree [K : Q].
  • E9.8: Let L/F be a field extension and let K := {a ∈ L | a is algebraic over F}. Show that if L is algebraically closed, then K is algebraically closed.
  • E9.9: Let F be a field of characteristic p > 0 satisfying the property that there is an element b ∈ F such that b is not a p-th power of an element of F. Show that there exists an irreducible inseparable polynomial q ∈ F[x].
  • E9.10: Let p be a prime number.
    (a) Let F be any field of characteristic p. Show that for all n ≥ 1 the function h_n: F → F defined by h_n(a) := a^pn (for all a ∈ F) is a ring homomorphism.
    (b) Let n be a positive integer, let q = x^pn - x ∈ Z/pZ[x], and let K be the splitting field of q over Z/pZ.
    (b-i) Show that the subset E ⊆ K consisting of all roots of q in K is a subfield of K.
    (b-ii) Show that |E| = pⁿ and E = K.
    (b-iii) Let L be any field with |L| = pⁿ. Let F' be the prime field of L. Show that F' ≅ Z/pZ and L is the splitting field of the polynomial q' = x^pn - x over F'. Show that this proves that any two fields of order pⁿ are isomorphic (as fields).
    (c) L be the field with |L| = pⁿ, and let h: L → L be the map defined by h(a) = a^p (for all a ∈ L). Show that hⁿ = Id_L, but h^m ≠ Id_L for all 1 ≤ m ≤ n-1.
    [Hint in (b-c): You're likely to use results from Math 817 on the relationship between the order of a group and the orders of elements in the group.]
  Due 4/11/18 for grading: E9.5(b,c-i), E9.7, E9.10(b)

• Problem Set 11:
  • E9.11: Let K be the splitting field of x⁶ - 4 over Q (from E9.7).
    (a) Give an explicit basis of K as a vector space over Q.
    (b) Let ζ = e^2πi/6 and let b be the unique positive real root of x⁶ - 4. Let g ∈ Aut(K/Q) be the automorphism that maps b to bζ², maps bζ² to b, and fixes all of the other roots of x⁶ - 4. For each k ∈ K, describe k and g(k) in terms of your basis for K.
    (c) Let h ∈ Aut(K/Q) be the restriction of the complex conjugation map to K. Determine the subfield K^{⟨ h ⟩} explicitly.
    (d) Determine whether there exists an element g' ∈ Aut(K/Q) that satisfies g'(b) = bζ.
  • E9.12: This problem compares Galois groups of extensions by n-th roots (of elements of the base field) depending upon whether the base field contains the n-th roots of 1.
    (Note: Since parts of this problem ask you to prove parts of TOC Props 9.107 and 9.110, you may not use those results in your proofs for this problem.)
    (a) Let p be a prime number.
    (a-i) Determine the elements of the Galois group G of x^p - 2 over Q. (That is, G = Gal(L/Q) where L is the splitting field of x^p - 2 over Q.)
    (a-ii) Is G abelian?
    (b) Let F be a field and let n be a positive integer such that char(F) does not divide n. A primitive n-th root of 1 over F is an element ζ in the splitting field K of xⁿ - 1 over F such that ζ generates the (multiplicative) subgroup H := {roots of xⁿ - 1 in K} ≤ K^x.
    (b-i) Show that |H| = n (that is, show that 1, ζ, ζ²,...,ζ^n-1 are distinct).
    (b-ii) Let b ∈ F, and let L be the splitting field of xⁿ - b over F. Show that if F contains a primitive n-th root of 1, then L/F is a finite Galois extension and Gal(L/F) is isomorphic to a subgroup of Z/nZ and hence is cyclic.
  • E9.13: Show that Q(√(2+√(2))) / Q is a Galois extension of degree 4 with Galois group that is a cyclic group of order 4.
  • E9.14: Let q = x⁴ -2x² -2 ∈ Q[x].
    (a) Show that q is irreducible in Q[x].
    (b) Show that the roots of q are b₁ = √(1 + √(3)), b₂ = √(1 - √(3)), b₃ = -√(1 + √(3)), and b₄ = -√(1 - √(3)).
    (c) Let K₁ := Q(b₁), K₂ := Q(b₂), and F := Q(√(3)).
    (c-i) Show that K₁ ≠ K₂, and K₁ ∩ K₂ = F.
    (c-ii) Prove that K₁, K₂, and K₁K₂ are Galois over F.
    (c-iii) Let G := Gal(K₁K₂ / F). Show that G is isomorphic to C₂ × C₂, and write out (explicitly, using the number of the roots from part (b)) the images of the elements of G under the embedding G → S₄ of TOC Thm 9.78. Determine all of the subgroups H of G and determine their corresponding fixed subfields (K₁K₂)^H.
    (d) Prove that the splitting field L of q over Q satisfies [L : Q] = 8, and Gal(L/Q) is isomorphic to the dihedral group of order 8.
  • E9.15: Let F be a field such that either char(F) = 0, or else char(F) = p for a prime number p and F = F^p (that is, every element of F is a p-th power of an element of F). Prove that if K is the splitting field over F of a (not necessarily separable) polynomial in F[x], then K/F is a Galois extension.
  • E9.16: Let K/F be a finite Galois field extension, and let b ∈ K. Let {b₁,...,b_m} be the orbit of b under the action of Gal(K/F) (with b₁ := b and b_i ≠ b_j whenever i ≠ j). Prove that m_b,F = (x-b₁) ··· (x-b_m).
  • E9.17: Let F be a field, let q ∈ F[x] be a separable irreducible polynomial of degree n, let L be the splitting field of q over G, and let b be a root of q in L. Let K be a Galois extension of F contained in L, and define m := [F(b) ∩ K : F] and d := [K(b) : K]. Show that the polynomial q splits into a product of m irreducible polynomials each of degree d over K.
    (Hint: This is exercise 14.2.28 on page 584 of the text; a hint for the proof is given there.)
  Due 4/20/18 for grading: E9.12(a-i,b-ii), E9.14(c-ii,c-iii), E9.15

S. Hermiller.