Math 818  Spring 2018  Problem Sets
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 C_{3} = ⟨ a  a^{3} = 1 ⟩
and let R be the group ring (Z/2Z)C_{3}.
(a) Let S = (Z/2Z)[y]/(y^{3}1) and
T = Z[x]/(2,x^{3}1).
(ai) Show that R and S are isomorphic rings.
(aii) 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+a^{2}).
 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=0}^{n} b(n,i)r^{i}s^{ni}
for all r,s ∈ R and n ≥ 0, where
b(n,i) is the sum 1 + 1 + ··· + 1 of
(n!)/(i!)((ni)!) 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  r^{n} ∈ 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).
(ci) Show that if R/N(R) is a field, then every element of R
is either nilpotent or a unit.
(cii) 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^{1}R is isomorphic to
a subring of the field of fractions of R (and hence S^{1}R
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^{1}R where S = {r^{n}  n ≥ 0}.
Prove that R[1/r] ≅ R[x]/(rx1).
Due 1/19/18 for grading: E5.10(ai,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]/(x^{2}+2).
Show that R ≅ S.
(b) Let T be the subring of M_{2}(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.
(di) Recall that the restriction to R of the complex norm,
N: R → Z (where
N(a +b√(2)) = a^{2} + 2b^{2}) 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.)
(dii) 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 I_{1} ⊊ I_{2} ⊊ I_{3} ⊊ ...;
that is, if the strict partial order is wellfounded.
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,dii), 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.
(bi) 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'.
(bii) 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 xa is p(a), and
consequently xa divides p if and only if a is a root of p.
(c) Suppose that the degree of the polynomial p is n ≥ 1.
(ci) Show that for all q ∈ F[x] there is a q' ∈ F[x] such that
degree(q') ≤ n1 and q + (p) = q' + (p).
(cii) Show that if q,r ∈ F[x] are polynomials of degree ≤ n1
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,x^{3}+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 x^{2}+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 = x^{2} + y^{2}  1.
(b) R = Q[x] and p = x^{4} + x + 1.
(c) R = Z/2Z[x] and p = x^{5} + 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 = x^{2} + i.
 E6.13:
(a) Show that (Z/2Z)[x]/(x^{2}+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 p^{2}.
 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 degreelex order:
x^{a}y^{b} > x^{c}y^{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:
(ai) I = (x^{2} + y^{2} + 1, x  y)
(aii) J = (x^{3}  y, x^{2}y  x)
(b) Use your Groebner basis for the ideal J in (aii) to
determine whether or not each of the following polynomials is in J:
p = x^{2}y  y^{3},
q = x^{5}y^{2}  x^{4}y + x^{3}  y^{2}
(c) Show that the Groebner basis for the ideal
K = (x^{2}+x+1, x+1) contains the rewriting rule 1 → 0.
Use your proof to find r,s ∈ R such that
1 = r(x^{2} + x + 1) + s(x + 1).
Due 2/9/18 for grading: E6.11, E6.12(a,b), E6.13(d), E6.16(aii,b)
 Problem Set 5:
 E7.1/E6.17: Let R be a commutative ring with 1 ≠ 0.
For any Rmodules M,N in this problem, let Hom_{R}(M,N)
have the Rmodule structure in TOC Prop 7.12. Also all instances
of ≅ in this problem denote Rmodule isomorphisms.
Let A,B,M,N be Rmodules.
(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}(R^{n},N) ≅ Π_{i=1}^{n} 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 Rsubmodule 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 Rmodule.
(a) Show that the annihilator ann_{R}(N) =
{r ∈ R  rn = 0 for all n ∈ N} is a (2sided) 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_{n}x^{n} + ··· +
r_{0})v = r_{n}a^{n}(v) + ··· +
r_{0}v for all
r_{n}x^{n} + ··· +
r_{0} ∈ 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 Rmodule N is irreducible if N ≠ 0 and the only submodules of N
are 0 and N. An Rmodule N has length k if there is a chain of
submodules 0 = N_{0} ⊂ N_{1} ⊂ ···
⊂ 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 Rmodule is welldefined.)
Let R be a commutative ring with 1 ≠ 0 and let M,N be Rmodules.
(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 Rmodules).
(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 Rmodule, and let N be an Rsubmodule of M.
(a) Show that if M is finitely generated as an Rmodule,
then so is M/N.
(b) Show that if N and M/N are finitely generated as Rmodules,
then so is M.
 EE7.6/6.22: Let R be a commutative ring with 1 ≠ 0.
For any ideal J of R and Rmodule N, define
JN = {∑_{k=1}^{n} j_{k}n_{k} 
n ≥ 0, j ∈ J, n ∈ N}.
Let I be an ideal of R, and let M be an Rmodule.
(a) Show that IM is an Rsubmodule 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 R^{n}/IR^{n}
≅ (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} ≅ F^{n} (as Fvector spaces) if and only if
m = n. Show that R^{m} ≅ R^{n} (as Rmodules) 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 Rmodule for
all α in an index set J, then the direct sum
⊕_{α ∈ J} M_{α} is
a free Rmodule.
(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 λ_{1},...,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 {v_{1},...,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 Zmodule, and hence Q is not a
free Zmodule. [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 Rsubmodules of an Rmodule L, and let P be the
Rsubmodule of L generated by M ∪ N.
(a) Show that P = M + N.
(b) Show that the
following are equivalent:
(bi) The function f: M ⊕ N → M + N defined by
f(m,n) = m + n is an Rmodule isomorphism.
(bii) M ∩ N = {0}. (biii) 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: (ci)
dim(V ⊕ W) = dim(V) + dim(W). (cii) 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,x^{2},...]
Due 2/23/18 for grading: E6.23(a), E6.24(b), E8.1(b; equivalence of
bi and bii 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: F^{n} → F^{m} and let
B = {b_{1},...,bn}
and C = {c_{1},...,cm}
be the standard ordered bases of F^{n} 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 ith row
of A' is 1, all other entries in the j_{i}th column are 0, and
j_{1} < j_{2} < ··· j_{r},
where r is the number of nonzero rows of A'.
Show that a basis for the image f(F^{n}) 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 Fvector 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.
(bi) 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.)
(bii) 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 welldefined (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_{n}x^{n} + ···
+ a_{0} ∈ F[x], let p(A) denote the matrix
p(A) = a_{n}A^{n} + ···
+ a_{0}I_{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 ijth 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 CayleyHamilton 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
CayleyHamilton 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 CayleyHamilton 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 Rmodule.
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
torsionfree if Tor(M) = 0.
(a) Show that if I is a nonprincipal ideal of R,
then I is a torsionfree Rmodule that is not a free Rmodule.
(b) Show that if M and N are Rmodules, then Tor(M ⊕ N) = Tor(M) ⊕ Tor(N).
(c) Suppose that R is a PID, and that M is a finitely generated
Rmodule. Show that if M is a torsionfree Rmodule then M is a free Rmodule.
 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 Rmodule 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, nonredundant list of representatives
of the conjugacy classes of GL_{3}(Z/2Z).
 E8.13: (a) Show that there does not exist a matrix
A ∈ Mat_{3,3}(Q) satisfying A^{8} = I_{3}
but A^{4} ≠ I_{3}.
(b) Give an example of a matrix B ∈ Mat_{4,4}(Q)
satisfying B^{8} = I_{4}
but B^{4} ≠ I_{4}.
 E8.14: Let V = R^{3} 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 = {b_{1},...,b_{n}},
let t: V → V be a linear transformation, let A = [t]_{B}^{B},
let E = {e_{1},...,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 Fvector 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} = x^{n} + b_{n1}x^{n1} +
··· b_{0} with each b_{i} ∈ F.
(a) Show that b_{0} = (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.
(bi) Show that if A' is similar to A, then tr(A') = tr(A).
(bii) Show that b_{n1} = 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} = (xλ_{1})
··· (xλ_{k}) for distinct
elements λ_{1},...,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.
(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 = F^{n} have the standard basis E = {e_{1},...,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 x^{3}  p is
irreducible over F.
 E9.3: Let K/F be a field extension, and let
a_{1},...,a_{m} ∈ K.
(a) Let L_{0} := F and for each 1 ≤ i ≤ m let
L_{i} := L_{i1}(a_{i}). Show that
F(a_{1},...,a_{m}) = L_{m}.
(b)
For a field K with subfields K_{1},...,K_{m},
the composite field of K_{1},...,K_{m},
denoted K_{1} ··· K_{m},
is the smallest subfield of K containing K_{1},...,K_{m}.
Show that F(a_{1},...,a_{m}) =
F(a_{1}) ··· 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
Flinear 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.
(ci) Show that N_{K/F} induces a
group homomorphism K^{x} → F^{x} and that
N_{K/F}(a) = a^{n} for all a ∈ F.
(cii) Suppose further that K = F(b) and
m_{b,F} = x^{n} + a_{n1}x^{n1} + ···
+ a_{0} is the minimum polynomial for b over F (with each a_{i}
∈ F). Show that N_{K/F}(b) = (1)^{n}a_{0}.
 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
x^{6}  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 pth 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.
(bi) Show that the subset E ⊆ K consisting of all
roots of q in K is a subfield of K.
(bii) Show that E = p^{n} and E = K.
(biii) Let L be any field with L = p^{n}.
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 p^{n}
are isomorphic (as fields).
(c) L be the field with L = p^{n},
and let h: L → L be the map defined by h(a) = a^{p}
(for all a ∈ L). Show that h^{n} = Id_{L}, but
h^{m} ≠ Id_{L} for all 1 ≤ m ≤ n1.
[Hint in (bc): 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,ci), E9.7, E9.10(b)
 Problem Set 11:
 E9.11:
Let K be the splitting field of x^{6}  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 x^{6}  4.
Let g ∈ Aut(K/Q) be the automorphism that
maps b to bζ^{2}, maps bζ^{2} to b, and fixes all of the other
roots of x^{6}  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 nth roots (of elements of the base field)
depending upon whether the base field contains the nth 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.
(ai) 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.)
(aii) 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 nth root of 1 over F is an element ζ in
the splitting field K of x^{n}  1 over F such that
ζ generates the (multiplicative) subgroup
H := {roots of x^{n}  1 in K} ≤ K^{x}.
(bi) Show that H = n (that is, show that
1, ζ, ζ^{2},...,ζ^{n1}
are distinct).
(bii) Let b ∈ F, and let L be the splitting field of
x^{n}  b over F.
Show that if F contains a primitive nth 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 = x^{4} 2x^{2} 2 ∈ Q[x].
(a) Show that q is irreducible in Q[x].
(b) Show that the roots of q are
b_{1} = √(1 + √(3)),
b_{2} = √(1  √(3)),
b_{3} = √(1 + √(3)), and
b_{4} = √(1  √(3)).
(c) Let K_{1} := Q(b_{1}),
K_{2} := Q(b_{2}),
and F := Q(√(3)).
(ci) Show that
K_{1} ≠ K_{2}, and K_{1} ∩ K_{2} = F.
(cii) Prove that K_{1}, K_{2}, and K_{1}K_{2}
are Galois over F.
(ciii) Let G := Gal(K_{1}K_{2} / F).
Show that G
is isomorphic to C_{2} × C_{2}, and write out
(explicitly, using the number of the roots from part (b)) the
images of the elements of G under the embedding
G → S_{4}
of TOC Thm 9.78.
Determine all of the subgroups H of G and determine their corresponding
fixed subfields (K_{1}K_{2})^{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 pth 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 {b_{1},...,b_{m}} be the orbit of b under the action
of Gal(K/F) (with b_{1} := b and b_{i} ≠ b_{j} whenever
i ≠ j). Prove that m_{b,F} = (xb_{1}) ···
(xb_{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(ai,bii), E9.14(cii,ciii), E9.15
S. Hermiller.