MATH 2510 ALGEBRA FALL 2024
ALL PROBLEM SETS
Last updated December 12, 2024
Please do NOT write your name on your problem sets. They’ll be graded anonymously.
Do, however, credit anyone you worked with on the problem set. Write their names
on a separate page at the end of your problem set, which the grader will not read.
Provide proofs for all claims, unless otherwise indicated.
Problem Set 1. Due Friday September 20 at 8pm Eastern Time
Please submit this assignment by uploading a PDF under “Assignments” on Canvas.
1. Problem I.3.1 (on the opposite category)
2. Problem I.4.2 (on which binary relations give rise to groupoids)
3. Problem I.5.12 (on fibered products and coproducts)
4. Let C be any category, X be an object, and F a final object. Does the product X × F exist?
5. Prove that an inclusion of a dense subspace in a Hausdorff topological space is an epimorphism
in the category of Hausdorff topological spaces.
6. For each of the following categories, determine the initial object, if one exists. In this problem
only, just the answers are fine in lieu of full proofs, as long as the answers are correct!
(a) The category of fields.
(b) The coslice category C X , for any category C and any object X.
(c) (extra credit) First, let FdVectk denote the groupoid of finite dimensional k-vector
spaces, with morphisms being invertible linear maps.
Next, given any set C, we may regard C as a category which we shall denote C whose
objects are the elements of C and with no nonidentity morphisms.
F
→ C for any set C, and
Now consider the category whose objects are functors FdVectk −
whose morphisms are commuting triangles
FdVectk
F′
F
C
φ
1
/ C′
Problem Set 2. Due Friday September 20 at 8pm Eastern Time
Please submit Question 1 under “Discussions” on Canvas, Question 2 by e-mail to me, and the
rest of the assignment by uploading a PDF under “Assignments” on Canvas.
1. Please introduce yourself under the discussion “Introductions” on the Discussion tab on Canvas. More specific guidelines are there.
2. To inform my planning, please send me an email with the following information:
(a) I am planning on assigning stable-ish groups for doing problem sets for those who opt
in. Would you like to be assigned a group? Is there anything I should know about group
work for you—what you like and don’t like? If there is anyone you’d like to work with,
let me know. In rare occasions, if there is someone you think I specifically should not
assign you to work with, please let me know.
(b) Anything else you want to tell me or ask me?
3. Ungraded: Read sections II §1,2, and 4 of the textbook.
4. Problem II.2.4 on the dihedral group. No proof needed, just the answer.
5. Problem II.3.5 on whether Q is a direct product
6. Problem II.4.3 on groups of order n having an element of order n
7. Problem II.4.8 on inner automorphisms
8. Problem II.4.14 on the size of Aut(Z/nZ)
9. Problem II.7.1 on subgroups of S3 . No proof needed, just the answer.
10. Problem II.7.7 on normality of subgroup generated by order n elements
11. Let A be any set. Is there a “free group with trivial squares” on A? Precisely, is there a
group F ts (A), and a map of sets f : A → F ts (A) such that f (a)2 = e for all a ∈ A, satisfying
the universal property that if G is any group and g : A → G is any map of sets such that
g(a)2 = eG for all a ∈ A, there is a unique group homomorphism F ts (A) → G making the
triangle below commute?
/ F ts (A)
A
G
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Problem Set 3. Due Friday September 27 at 8pm Eastern Time
1. Read the introductions on Canvas under “Discussions,” and leave some replies in that thread.
2. Understand the statements and proofs of Propositions II.8.9, II.8.10, II.8.11 (isomorphism
theorems). Now close your book. Write down the theorem statements in your own words
from memory. Pick one of those theorems and imagine you are teaching it; write down an
interesting example that you think illustrates the theorem well.
3. Problem II.2.5 on a presentation of dihedral groups
4. Problem II.6.1 on matrix groups. (Try to think of these groups as geometrically as possible.)
5. Problem II.8.17: if G is a finite abelian group with order a multiple of p, then G has an
element of order p.
6. For H a subgroup of G, we noted that G/H is naturally an object in the category of G-sets.
Prove that
AutG-Set (G/H) ∼
= NG (H)/H,
where NG (H) denotes the normalizer of H in G.
As a special case, figure out how many automorphisms a G-torsor has, for a finite group
G. Do you think your answer aligns with the slogan “a torsor is a group that has lost its
identity?”
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Problem Set 4. Due Friday October 4 at 11:59pm Eastern Time
1. Let n ≥ 1 be an integer, and let λ = (λ1 , . . . , λk ) be a partition of n with λ1 ≥ · · · ≥ λk . For
example, λ = (5, 5, 4, 2) is a partition of 16. Let σ be an element of Sn with cycle type λ.
(a) What is the size of the centralizer ZSn (σ) of σ?
(b) What is the size of the normalizer NSn (⟨σ⟩) of the subgroup generated by σ?
2. Let G be the group of invertible n × n matrices over the field Fp , for p prime.
(a) Show that the subgroup of upper triangular matrices with all ones on the diagonal is a
Sylow p-subgroup.
(b) How many Sylow p-subgroups does G have?
3. Let G be a group. On Monday we will give the example of the wreath product G wr Sn :=
Gn ⋊ϕ Sn where ϕ : Sn → AutGrp (Gn ) is given for each σ ∈ Sn by
ϕσ ((g1 , . . . , gn )) = (gσ−1 (1) , . . . , gσ−1 (n) ).
Suppose G acts on a set A. Construct an interesting action of G wr Sn on the set A×{1, . . . , n}.
4. For each n ≥ 1, argue whether GLn (R) is the (direct product / semidirect product / neither
of these) of its subgroup SLn (R) with some other subgroup of GLn (R).
5. For any group G, the group G/[G, G] is called its abelianization. Practice your universal
properties by proving, using only universal properties,1 that the abelianization of a free group
is a free abelian group. That is, for any set A, prove using only universal properties that
F (A)/[F (A), F (A)] ∼
= F ab (A).
1
In other words, your proof should not reference any actual constructions of (free groups / free abelian groups /
abelianizations), but only invoke the universal properties that we already know they satisfy.
4
Problem Set 5. Due Friday October 11 at 8pm Eastern Time
Ungraded: study the classification of finite abelian groups, i.e., Theorem IV.6.6 on p. 237. Be
willing to invoke this theorem.
1. Let n = (p1 · · · pr )2 , for p1 , . . . , pr distinct prime numbers. How many isomorphism classes of
abelian groups of order n are there?
2. Classify groups of order 21. That is, produce (with proof) a list of groups such that every
group of order 21 is isomorphic to exactly one group on the list. Also indicate, for each group
on your list, how many elements of each order there are.
3. Let B be the group2 of invertible upper triangular n×n matrices over a field k. Is B solvable?
4. Problem IV.3.10 on nilpotent groups.
5. Let U be the group of upper triangular n × n matrices with 1 on the diagonal, over a field k.
Is U nilpotent?
6. Consider the following functors from a category to the category Set. Identify left adjoints to
these functors, or assert that none exists. Just the answers are fine, in lieu of proofs,
since the main purpose of this problem is sharpening intuition.
(a) The forgetful functor from the category of abelian groups to Set.
(b) The forgetful functor from the category of topological spaces to Set.
(c) Fix a prime p. The forgetful functor from the category of fields of characteristic p to
Set.
(d) Let A any set. The functor hA : Setop → Set sending a set S to MorSet (S, A).
2
B stands for Borel.
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Problem Set 6. Due Friday October 18 at 8pm Eastern Time
Ungraded: Study the definition of the product R1 ×R2 of rings. (Either study it from the textbook,
or guess it yourself and prove that your guess is correct by checking the universal property of
products.)
1. Let R be a ring. Define an R-module structure on R. Now suppose that R is a commutative
ring. Prove, or disprove, that every surjective R-module morphism R → R is injective.
2. Let R be a ring. An idempotent in R is an element e satisfying e2 = e. Prove that a
commutative ring R is isomorphic to a product of two nonzero rings if and only if R has an
idempotent other than 0R , 1R .
3. Let k be a field. To warm up, prove that the units of k[[x]] are exactly the power series with
nonzero constant term. Now classify the ideals of k[[x]]. What about the ideals of k((x))?
4. Let I : D → C be the inclusion functor for a full subcategory3 D of C. Suppose I has a left
adjoint F : C → D. For each object X in C, let fX : X → F X denote the image of 1F X under
the bijection
∼
=
MorD (F X, F X) −
→ MorC (X, F X)
provided by adjunction. Prove that if fX is monic for each X in C, then fX is epic for each
X in C.
5. To make the previous problem more concrete:4 let C be the category of integral domains; the
morphisms in C are injective homomorphisms. Find an appropriate subcategory D and use
Problem 4 to (re)derive the fact that R → Frac R is an epimorphism in C, for any integral
domain R.
3
4
See Exercise I.3.8 on page 26.
A sketch of an argument is fine here; this problem is mainly meant to guide intuition.
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Problem Set 7. Due Friday October 25 at 8pm Eastern Time
Ungraded:
• Look over submodules/quotients/isomorphism theorems for modules, pp. 160–162.
• Study products/coproducts of modules, pp. 164–166.
Let A be an abelian group. An A-grading on a commutative ring R is a direct sum5 decomposition
M
R=
Ra
a∈A
of abelian groups, such that Ra Rb ⊆ Ra+bP
. The elements of Ra are called homogeneous of degree
a. An element r ∈ R can be written r = a∈A ra with ra ∈ Ra ; we call the ra the homogeneous
components of r. An ideal I ⊆ R is called homogeneous if it is generated by homogeneous elements.6
1. Let A be an abelian group and let R be an A-graded commutative ring.
(a) Prove that I is homogeneous if and only if for every element r ∈ I, the homogeneous
components of r are in I.
(b) If I is a homogeneous ideal, show that R/I inherits an A-grading from that on R.
(c) Let I and J be homogeneous ideals. Are the ideals I +J, I ∩J, and (I : J) homogeneous?
2. Let n ≥ 1 be an integer, let k be a field; we shall consider ideals in the ring R = k[x1 , . . . , xn ].
For s = (s1 , . . . , sn ) ∈ (Z≥0 )n , write xs = xs11 · · · xsnn for short. Give a (finite) algorithm that
takes as input two finite subsets A and B of (Z≥0 )n and outputs a finite set C ⊂ (Z≥0 )n with
⟨xa : a ∈ A⟩ ∩ ⟨xb : b ∈ B⟩ = ⟨xc : c ∈ C⟩.
3. Let R be an integral domain, and suppose S ⊂ R is a multiplicative subset such that S −1 R
is a field. Must we have S −1 R ∼
= Frac R?
4. Let S ⊂ R be a multiplicative subset of a commutative ring R, and let ϕ : R → S −1 R denote
the localization map. Is it ever the case that the ideals of S −1 R are in bijection with the
ideals of R, via contraction—other than the case that ϕ is an isomorphism?
5. Let R be a commutative ring, and let M, N1 , and N2 be R-modules. Suppose
f1 : N1 → M,
f2 : N2 → M
are R-linear maps such that f1 (N1 ) + f2 (N2 ) = M . Prove that if N1 and N2 are Noetherian,
then so is M .
5
6
The coproduct in the category of abelian groups or modules over a ring is often called the direct sum.
The most common example is A = Z. A good example is the Z-grading on k[x1 , . . . , xn ] by total degree.
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Problem Set 8. Due Friday November 1 at 8pm Eastern Time
1. Problem V.1.7, proving a version of the Hilbert Basis Theorem for rings of formal power
series.
√
2. Problem V.1.17, studying the ring Z[ −5] and showing it is not a UFD.
3. (moved to extra credit; full credit given for blank solutions) Let k be a field. Let
f ∈ k[x, y] be a nonzero polynomial that lies in the ideal (x, y)2 . Prove that k[x, y]/(f ) is not
a UFD.7
4. A valued field is a field K together with a function v : K ∗ → Γ, for (Γ, ≤) a totally ordered
abelian group, satisfying
• v(ab) = v(a) + v(b) for a, b ∈ K ∗ ,
• v(a + b) ≥ min(v(a), v(b)) for a, b ∈ K ∗ such that a + b ∈ K ∗ .
The function v is called a valuation. The valuation ring of K is OK = {0} ∪ {x ∈ K ∗ :
val(x) ≥ 0}; you may wish to check that this is a ring.
(a) Prove that if a, b ∈ K ∗ with v(a) ̸= v(b), then a + b ̸= 0 and v(a + b) = min(v(a), v(b)).
(b) Prove that a valuation ring OK is local, meaning that it has a unique maximal ideal mK .
Its residue field is defined to be the quotient k = OK /mK .
(c) If (K, v) is a valued field with v : K ∗ → Z surjective, then K is called a discretely valued
field and OK is called a discrete valuation ring. Prove that discrete valuation rings are
Euclidean domains.
(d) Two familiar examples of valued fields are F ((x)) for F a field, with valuation being
lowest exponent; and (fixing any prime p) Q with its p-adic valuation v(pn · ab ) = n
for p not dividing a, b. Briefly check that the claimed functions are indeed valuations
on these two fields. Now identify the valuation rings and residue fields of these two
discretely valued fields. Conclude from these two examples that the characteristic of K
and that of its residue field k may or may not agree.
7
In algebraic geometry, you will learn that the failure of this ring to be a UFD is related to the fact that the zero
locus of f is a plane curve with a singularity at the origin.
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Problem Set 9. Due Friday November 8 at 8pm Eastern Time
Ungraded: study pp. 288-289 on Eisenstein’s criterion.
Recall that by abuse of notation, quantities like gcd and content are well-defined only up to multiplication by units, and equalities may be interpreted as equality up to multiplication by a unit.
1. (a) Let K be a discretely valued field, with valuation ring OK . Prove that Frac OK ∼
= K.
(b) If R is a commutative ring, and p is any prime ideal of R, write S = R \ p; then we write
Rp := S −1 R. Prove that if R is a PID then Rp is a DVR.8
2. Is content well-defined under automorphisms of R[x] that fix R? Precisely: let R be a UFD,
let f ∈ R[x] a nonzero polynomial, and let ϕ : R[x] → R[x] be an automorphism of the ring
R[x] that fixes the subring R. Must we have contϕ(f ) = contf ?
3. Let R be a UFD and let f ∈ R[x, y] be nonzero. Let q ∈ R[x] be the content of f , regarding
f as as a polynomial in y with coefficients in R[x]. Let r ∈ R be the content of q.
Prove or disprove: r is a greatest common divisor of the coefficients of f .
4. Problem V.5.24, proving C[x, y, z, w]/(xw−yz) is an integral domain by Eisenstein’s criterion.
5. Let k be a field. Define a category C whose objects are pairs (V, α) for V a k-vector space
and α : V → V an invertible linear map. (What are the morphisms in C?) Now find, with
proof, a principal ideal domain R such that the category of R-modules is equivalent to C.
8
A possible hint is to imitate the example of the p-adic valuation on Q appearing in the previous problem set.
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Problem Set 10. Due Friday November 15 at 8pm Eastern Time
Ungraded: study Smith normal form in section VI.2.4, pp. 323–324.
1. Suppose the minimal polynomial of a linear map L : Cn → Cn is tm , for some 0 < m ≤ n.
What are the possibilities for the rank of L?
2. Prove the genuinely useful fact that commuting diagonalizable linear maps are simultaneously
diagonalizable.
More precisely, let V be a finite dimensional vector space over k, and let α1 , . . . , αm ∈ Endk (V )
be diagonalizable linear maps such that αi αj = αj αi for all i and j. Show that α1 , . . . , αm are
simultaenously diagonalizable, i.e., show that V has a basis consisting common eigenvectors
of α1 , . . . , αm .
The following lemmas might be helpful:
(a) Exercise VI.7.12.
(b) Prove that the eigenspaces of α1 are invariant subspaces under α2 , . . . , αm . Apply Exercise VI.7.12 and induct on m.
3. For a connected graph G with vertex set {1, . . . , n}, the Laplacian ∆(G) is the n × n matrix
given by
(
(−1) · (number of edges incident to vertex i)
if i = j
∆(G)i,j =
number of edges between vertex i and vertex j if i ̸= j.
e
The reduced Laplacian ∆(G)
is the (n − 1) × (n − 1) matrix obtained by deleting row k and
column k, for some (any) choice of k ∈ {1, . . . , n}. View the reduced Laplacian as a map
e
e
∆(G)
: Zn−1 → Zn−1 . Then the sandpile group SG is defined to be the cokernel of ∆(G);
SG
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is a finite abelian group.
Let Kn denote the complete graph on n vertices.
(a) For n = 3, 4, 5, compute SKn in its invariant factor form, by computing by hand the
e n ). Make a conjecture characterizing SKn for all n.
Smith normal form of ∆(K
(b) Prove your conjecture. Some possible lemmas that could be helpful:
i. Let Aa,n denote the (n + 1) × (n + 1) square matrix with as on the diagonal and 1s
off the diagonal. Argue briefly that det(Aa,n ) = (a + n)(a − 1)n , e.g., by subtracting
one row from all others.
ii. Show that, in general, the product d1 · · · dk of the first k invariant factors of the
Smith normal form of a matrix A is the gcd of all the k × k minors of A.
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There are interesting theorems on the distribution of sandpile groups, and in particular their Sylow subgroups,
of Erdős–Renyi random graphs.
10
Problem Set 11. Due Wednesday November 27 at 8pm Eastern Time
Ungraded: Study Example VII.1.19 on pp. 395–396.
1. Let K = C((t)) and let α = t1/4 + t3/4 + t5/4 + t7/4 + · · · ∈ C{{t}}. What is the size of
AutK (K(α))?
2. Exercise VII.1.24.
3. Exercise VII.1.25.
4. Prove that the residue field of an algebraically closed valued field is algebraically closed.
5. Fix an integer d ≥ 1. Suppose k ⊂ L and k ⊂ F are field extensions. Suppose that every
polynomial of degree at most d in k[x] factors into linear factors over L. Suppose also that
every element of F is the root of a nonzero polynomial in k[x] of degree at most d.
Must there exist an embedding F ,→ L making the diagram below commute?
8 LO
k
&
11
F
Problem Set 12. Due Friday December 6 at 8pm Eastern Time
1. Exercise VII.4.2, describing the splitting fields of x6 + x3 + 1 and of x4 + 4 over Q. Please
give generators for these fields over Q, and calculate the degree of the field extension.
2. Exercise VII.4.5, clarified as follows: Let F be a splitting field for a polynomial f (x) ∈ k[x],
and let g(x) ∈ k[x] be a factor of f (x). Prove that there is a unique intermediate field
k ⊂ F ′ ⊂ F such that k ⊂ F ′ is a splitting field for g(x).
3. Exercise VII.4.7, on normal extensions.
4. In class we studied the field Fp (t) as an example of a field that is not perfect. Does every
field that is not perfect contain a subfield isomorphic to Fp (t) for some prime p?
5. (updated) Exercise VII.5.4, on counting irreducible polynomials over F2
6. It is interesting to count over finite fields. For example, for any positive integer n, let
[n]q := 1 + q + · · · + q n−1 ∈ Z[q],
[n]!q := [n]q · [n − 1]q · · · · · [1]q ∈ Z[q].
The polynomial [n]q is called the q-analog of n, since upon setting q = 1, we recover the
number n. Prove that the function
{prime powers} → Z≥0
sending q to the number of k-dimensional subspaces of Fnq agrees with
[n]!q
n
:=
k q
[k]!q [(n − k)]!q
as a function of q. In fact, explain why the expression above is actually a polynomial in q
with integer coefficients, in which the coefficient of q i is the number of locations of pivots in a
reduced row echelon forms of a k × n matrix that has k pivots and i undetermined entries.10
Stretch your imagination and explain why it might be reasonable to define a “k-dimensional
subspace of an n-dimensional vector space over F1 ” to be a k-element subset of {1, . . . , n}—
even in the absence of a definition of a field F1 with one element.
10
For example, this reduced row echelon form has three pivots (the 1s) and four undetermined entries (the ∗).


0 1 ∗ 0 0 ∗
0 0 0 1 0 ∗
0 0 0 0 1 ∗
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Problem Set 13. Optional and ungraded
1. Problems VII.6.4, VII.6.6, and VII.6.7.
Practice final exam. Optional and ungraded
1. What is a monomorphism in a category C? Which morphisms in the category of groups are
monomorphisms?
2. What is the size of the group GL3 (F2 ) of 3 × 3 invertible matrices over F2 ? Does there exist
a set of size 21 on which GL3 (F2 ) acts transitively?
3. Classify groups of order 15.
4. State the universal property of localization of commutative rings. Is it possible to localize a
commutative ring R that is not an integral domain and get an integral domain?
5. (typo corrected 12/12) Let R be a commutative ring. What does it mean for an R-module
to be Noetherian? Is a finitely generated R-module necessarily Noetherian?
6. Let R be an integral domain. Define irreducible element and unique factorization domain. Is
every integral domain a unique factorization domain?
7. Let n ≥ 1, and suppose A : Cn → Cn is an idempotent linear operator. (Idempotent means
that A2 = A.) What are the possibilities for the characteristic polynomial of A?
8. Give a complete proof that the multiplicative group of a finite field is cyclic. You may use the
structure theorem of finitely generated modules over a PID, as long as you state it correctly.
9. Let k be a field. What does it mean for a field k to be perfect? What does it mean for a
polynomial f (x) ∈ k[x] to be separable? Prove that if all irreducible polynomials in k[x] are
separable, then k is perfect.
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