Lecture and Homework Log
Math 401
Lecture 1
, August 25:
Introduction to the course, overview of the origins of algebra: factoring, partial fractions, integration, elliptic integrals, addition law, the group law on an elliptic curve. Read Chapter 0.
Lecture 2
, August 27:
Definition of group and group homomorphism, immediate consequences, examples from rings and fields, integers modulo n, integers modulo p. Definition of category, the category of sets, the category of real vector spaces, the category of groups.
Homework #1
1.1: 14, 18, 27, 35, 36, due Monday.
Lecture 3
, August 30:
Definition of isomorphism and automorphism in a category. Group of automorphisms of an object in a category. Examples: permutation groups, general linear groups, and their cardinality. Orthogonal matrices and the orthogonal group O(n). Symmetry group of a subset of Euclidean space. Dihedral group as the symmetry group of a regular n-gon, two of its elements and the equation they satisfy.
Lecture 4
, September 1:
Normal form for elements of the dihedral groups. Generators and relations for groups. Generators and relations for the dihedral groups. Permutations groups and cycle decomposition of permutations. The quaternion group as an example of a nonabelian group of order 8.
Homework #2
due Wednesday, September 8: 1.2: 17, 18; 1.3: 14, 19, and an extra homework problem: Construct a regular icosahedron by circumscribing it about 3 suitably chosen mutually perpendicular rectangles. Show it has 60 rotations of orders 1, 2, 3, and 5. (A rotation is an orthogonal matrix of determinant 1, and in dimension 3 it always is achieved by rotation about an axis through some angle.)
Lecture 5
, September 3:
Further clarification of the problem about the symmetries of the icosahedron. Homomorphisms and isomorphisms of groups. D
_{6}
is isomorphic to S
_{3}
. D
_{8}
embeds into S
_{4}
, etc. GL
_{2}
(F
_{2}
) is isomorphic to S
_{3}
. Subgroups and inclusion homomorphisms. D
_{2n}
as a subgroup of GL
_{2}
(R). Homomorphisms from a group defined by generators and relations to another group. A homomorphism from D
_{2n}
to D
_{2k}
when k divides n. Read chapter 2.
No school
, September 6.
Lecture 6
, September 8:
Starting chapter 3. Coimage and image in the category of sets. The coimage of a map f from X to Y is the set of (nonempty) fibers of f. The coimage and image are isomorphic. The coimage is a partition of X, and a quotient set of X. Partitions correspond to equivalence relations. Quotient by an equivalence relation. Action of a group G on a set X, on the left. The set of orbits G\X is a partition, and thus a quotient set of X. Actions on the right, with quotient X/G. Kernel K of a group homomorphism from G to H. Identity K\G = coim f = G/K. Left cosets, right cosets.
Lecture 7
, September 10:
Normal subgroups, quotient groups, quotient map, first isomorphism theorem, any subgroup of index two is normal, a normal subgroup of order 3 in S
_{3}
, splitting maps, Lagrange's Theorem, product groups, two groups of order 6, five groups of order 8, any group of prime order is cyclic.
Homework #3
due Friday, Sept 17: section 1.7: 21; section 3.1: 14, 15, 17, 21, 26, 32, 35, 36, 41.
Lecture 8
, September 13:
Section 4.1. The other isomorphism theorems for groups. Orbits of cyclic subgroups in S
_{n}
correspond to the disjoint cycles of the generator. Stabilizers and orbits: G/G
_{x}
=Gx, |G|/|G
_{x}
|=|Gx|. Examples. Additional problems for
Homework #3
due Friday: 4.1: 7, 10.
Lecture 9
, September 15:
The Hölder program: analyze a group G by analyzing K and G/K first, where K is a normal subgroup of G. Simple groups. Composition series for finite groups. Simple abelian groups are cyclic of prime order. Definition of solvable group. Composition series for cyclic groups amount to prime factorization of the order. Composition series for S
_{3}
. The Jordan-Hölder theorem and the start of the proof.
Lecture 10
, September 17:
The Jordan-Hölder theorem: end of the proof. Groups acting on themselves by conjugation: the orbits G*g are conjugacy classes, the stabilizers are centralizers C
_{G}
(g), we have a bijection G/C
_{G}
(g) = G*g, and an equality |G| / |C
_{G}
(g)| = |G*g|. A normal subgroup is a union of conjugacy classes. Conjugacy classes in S
_{n}
correspond to partitions of n.
Lecture 11
, September 20:
A formula for cardinality of a conjugacy class in S
_{n}
. The conjugacy classes of S
_{5}
: 120 = (1 + 15 + 20 + 24) + (10 + 30 + 20); the centralizer of (12)(34) is a D
_{8}
. The sign of a permutation defined: it's a homomorphism; calculating it from a cycle decomposition. The alternating group A
_{n}
, a normal subgroup of S
_{n}
of index 2. Conjugacy classes in A
_{n}
, according to whether the permutation commutes with an odd permutation, or whether the cycle decomposition consists of odd-length cycles of different lengths.
Lecture 12
, September 22:
Simplicity of A
_{5}
. Read the proof in the book, section 4.6, for the simplicity of A
_{n}
, for n greater than or equal to 5. Conjugates of stabilizers are stabilizers. The class equation; p-groups have non-trivial center, and thus are solvable (and nilpotent, but we didn't define that). Groups of order p
^{2}
are abelian. Cauchy's theorem: a group whose order is divisible by p has an element of order p.
Lecture 13
, September 24:
Introduction to rings. We will insist that rings have a 1. Examples: integers Z, rational numbers Q, real nubmers R, complex numbers C, quaternions H, endomorphism rings End(A) in categories where the Hom sets are abelian groups, matrix rings, polynomial rings, monoid rings, group rings. The group of units R
^{x}
. Ring homomorphisms. Examples, including a map from RQ
_{8}
to H. The kernel is an ideal, multiplication in the coimage.
Here
is a solution of problem 1.7: 21.
Homework #4
due Friday, October 1: 4.2: 7b, 10; 4.3: 30; and show that the group of rigid motions of the icosahedron (in SO(3)) is isomorphic to A
_{5}
.
Lecture 14
, September 27:
(Professor Griffith.) Temporarily, rings are assumed commutative. Ideals, quotient rings (factor rings), ring homomorphisms.
Lecture 15
, September 29:
(Professor Griffith.) Ring homomorphisms by substitution. Isomorphism theorems for rings. Zero divisors, integral domains, fields. Associate elements. Prime ideals, maximal ideals. Maximal ideals are prime. Prime ring elements, irreducible ring elements.
Lecture 16
, October 1:
(Professor Griffith.) Characterizing fields in terms of their ideals. Characterizing maximality or primeness of I in terms of R/I. Example of a chain of prime ideals in a polynomial ring over a field. Examples of maximal ideals in a polynomial ring over a field. Noetherian ring (ACC on ideals), and a characterization in terms of finite generation of ideals. PID's are noetherian. Nonzero prime ideals in a PID are maximal. Euclidean domains, and three examples.
Homework #5
due Monday, October 11: 8.1: 8a, 12; 8.2: 6; 8.3: 6.
Lecture 17
, October 4:
In an integral domain prime elements are irreducible. In a PID irreducible elements are prime. A Euclidean domain is a PID. Greatest common divisors, and the Euclidean algorithm; shortcuts available for integers or polynomials in one variable over a field.
Lecture 18
, October 6:
How to use blocks to embed permutation groups in smaller symmetric groups. Mentioned universal side divisors, see section 8.1 for an example of a PID which is not a Euclidean domain. Unique factorization domains (UFDs). A PID is a UFD.
Exam 1
, October 8:
Group theory. The scores were 16, 17, 20, 22, 25, 26, 30, 32, 32, 32, 34, B: 37, 37, 37, 37, 40, 40, 40, 41, 42, 42, 45, 45, 47, 48, 49, 49, 50, 50, 52, 52, 52, 52, 53, 53, 54, A: 60, 61, 62, 62, 70, 70, 72, 77, 88, and 96.
Lecture 19
, October 11:
A ring is noetherian if and only if any nonempty family of ideals has a maximal element. Construction of the field of fractions of an integral domain, universal property. Goal: if F is a field, then F[x
_{1}
,...x
_{n}
] is a UFD. There is an isomorphism between R[x,y] and R[x][y]. New goal: if R is a UFD, so is R[x]. If R is an integral domain, so is R[x]. From a homomorphism R -> S we get a homomorphism R[x] -> S[x]. Gauss' Lemma: if R is a UFD, F = frac(R), and p is in R[x], and p factors in F[x], then p factors in R[x], too, with factors differing by scalar multiples from the original ones.
Lecture 20
, October 13:
The proof of Gauss' lemma. If R is a UFD, so is R[x].
Lecture 21
, October 15:
Characteristic of a field. Field extensions, degree, finite extensions, simple extensions, primitive element, algebraic element, transcendental element, irreducible polynomial p(x). For an element b algebraic over F with irreducible polynomial p, we know that F(b)=F[b], and F[b] is isomorphic to F[x]/p.
Homework #6
due Friday, October 22: #1: How much of exam problem 2 goes through for 4 by 4 matrices? (You should find blocks of all possible sizes.); 8.2: 5; 9.1: 8, 13; 9.2: 4, 7, 10; 9.3: 1.
Lecture 22
, October 18:
Problem sessions:
Mondays at 7pm-9 in 218 MEB, Saturdays at 1pm-3 in 245 Altgeld, except 10/23 and 11/12 in 140 Burrill Hall. Any root b (in an extension field) of an irreducible polynomial p(x) over F gives an extension field F(b) isomorphic to F[x]/(p). Multiplicativity of degrees in a tower of field extensions. The degree over F of an algebraic element in K divides [K:F]. Finite extensions. A finite extension of a finite extension is finite. Algebraic extensions. Finite extensions are algebraic. The set of elements algebraic over F is closed under addition, subtraction, multiplication, and division. Straightedge and compass constructions -- no lecture, read about it in section 13.3. Definition of splitting field. Example: the splitting field of x
^{n}
-1 over the rational numbers is generated by exp(2 pi i / n).
Problem session 1
, October 18:
Is [Q(exp(2 pi i / 100)) : Q] = 100?
In each of the following problems, G acts on X on the left via an action g*x (to be defined in each example below) and you are to compute the quotient set G\X (set of orbits).
X = G, with g*x=gx.
X = GxG with g*(x,y)=(gx,gy).
S is a set, X = GxS with g*(x,s)=(gx,s).
G=Z
_{2}
generated by t, t*x=-x.
G=Z, X=R, g*x=g+x.
G=Z
^{2}
, X=R
^{2}
, g*x=g+x.
F=field, G=F
^{x}
, X=F
^{n}
, g*x=gx.
F=field, G=F
^{x}
, X=F
^{n}
- {0}, g*x=gx.
F=field, G=GL
_{n}
(F)xGL
_{m}
(F), X = Hom
_{F}
(F
^{m}
,F
^{n}
), (g,h)*x=gxh
^{-1}
.
Lecture 23
, October 20:
Existence and uniqueness of splitting fields. The degree of the splitting field of a polynomial of degree n is less than n!. Definition of derivative of a polynomial; the product rule. Definition of separable polynomial. Separable polynomials have roots of multiplicity one. The gcd of two polynomials doesn't change when the field is enlarged. A separable polynomial remains separable when the field is enlarged.
Lecture 24
, October 22:
Another proof that the gcd of two polynomials doesn't change when the field is enlarged. Every finite field is of cardinality q = p
^{n}
for some prime number p and some number n > 0. The splitting field L over Z/pZ of f(x) = x
^{q}
-x has cardinality q and consists entirely of the roots of f. Every finite field of cardinality q is isomorphic to L.
Homework #7
due Friday, October 29: 13.1: 1, 3, 5; 13.2: 2, 7; 13.3: 1; 13.4: 1.
Problem session 2
, October 23:
For subgroups K < H < G show [G:K]=[G:H][H:K]
For subgroups K < G > H with intersection L show [G:L] is less than or equal to [G:K][G:H].
For a subgroup H < G is |G/H|=|H\G|?
What are the commutator subgroups of SO
_{2}
and O
_{2}
?
Lecture 25
, October 25:
The Eisenstein criterion for irreducibility. The field Q(exp(2 pi i / p)) has degree p-1 over Q when p is a prime. Introduction to algebraic geometry and systems of equations. A hint about the Zariski topology.
Problem session 3
, October 25:
For subgroups K < G > H with intersection L show [G:L] is less than or equal to [G:K][G:H] with a more direct proof than last time.
What is the commutator subgroup of O
_{3}
?
What is the size of the conjugacy class of the matrix A in GL
_{2}
F
_{3}
of the linear map f defined by f(x,y) = (x,2y)?
Lecture 26
, October 27:
(Professor Griffith.) Hilbert basis theorem: if R is a commutative noetherian ring, so is R[x]. Description of maximal ideals in F[x
_{1}
,...,x
_{n}
], where F is a field.
Lecture 27
, October 29:
Definition of partially ordered set (poset), totally ordered set, chain, upper bound of a subset, least upper bound of a subset, uniqueness of least upper bounds, maximal elements. Zorn's lemma: a poset with an upper bound for any chain has a maximal element. A commutative ring is nonzero if and only if it has a maximal ideal. A commutative ring is nonzero if and only if it has a prime ideal. Definition of Spec(R). Definition of Zariski topology on F
^{n}
; a closed set is the solution set Z(S) associated to a set S of polynomials.
Homework #8
due Friday, November 5: 13.3: 4; 13.4: 2; 13.5: 2, 3, 6; 13.6: 1; 15.1: 2.
Problem session 4
, October 30:
Suppose G acts on a set X. How are the stabilizers of two points in the same orbit related?
Suppose H and K are conjugate subgroups of G. Are the G-sets G/H and G/K isomorphic? (A G-set is a set together with an action of G. The functions compatible with the action make the G-sets into a category.)
Suppose the G-sets G/H and G/K are isomorphic. Are H and K are conjugate?
Describe all S
_{3}
-sets of cardinality 4 up to isomorphism.
Prove the eigenvalues of a unitary matrix have absolute value 1.
Prove an orthogonal matrix in O(n) with n odd has a real eigenvalue, necessarily 1 or -1.
Prove any element of SO(3) has 1 as an eigenvalue, and thus is rotation about some axis.
Prove the commutator subgroup of O(3) is SO(3).
Prove the commutator subgroup of SO(3) is SO(3).
Lecture 28
, November 1:
If I is the ideal generated by S then Z(I)=Z(S). An algebraic set is defined by a finite number of equations. Definition of F-algebra and homomorphism of F-algebras. Algebraic sets of an ideal characterized in terms of the quotient ring R by homomorphisms of F-algebras from R to F. The subspace topology characterized in terms of the quotient ring. Effect of ring homomorphisms. Relationship with maximal ideals.
Problem session 5
, November 1:
Show that an m by n matrix of integers can be diagonalized by a sequence of elementary row and column operations of the type where an integer multiple of a row or column is added to another.
Prove the Chinese remainder theorem.
Prove that the product IJ of two ideals is equal to the intersection if I+J=R.
Compute Q[x]/(x
^{2}
-1).
Compute Q[x]/(x
^{2}
+1).
Compute Q[x]/(x
^{2}
). Show it is not a product of two nonzero rings.
Compute Q[x]/((x-1)
^{2}
).
Compute Z[x]/(x
^{2}
-1) following the ideas in the Chinese Remainder Theorem as far as possible.
Lecture 29
, November 3:
Hilbert Nullstellensatz for the complex numbers: Z(I) is nonempty iff I is a proper ideal. Why attempting to generalize to arbitrary commutative rings forces us to consider prime ideals as points.
Lecture 30
, November 5:
The spectrum of a commutative ring and the Zariski topology on it. A continuous map from Spec(R') to Spec(R) associated to a ring homomorphism from R to R'. Localizations - rings of fractions S
^{-1}
in general. The localization R
_{f}
; it's zero iff f is nilpotent. The nilpotent elements of a commutative ring form an ideal called the nilradical. A nilpotent element is contained in every prime ideal, and a nonnilpotent element is outside some prime ideal (using the existence of a maximal ideal in R
_{f}
). A handout (
dvi file
,
pdf file
, or
postscript file
) giving a short proof of Zorn's lemma.
Homework #9
due Friday, November 12: 15.1: 3; 15.2: 21; 15.4: 10; 15.5: 1, 14, 17. Here are the
solutions
.
Problem session 6
, November 6:
Let t = (1 + sqrt(-7))/2, and compute the ring Z[t]/(t
^{5}
-1).
What is the image of Spec(R/I) in Spec(R)?
What topological space is Spec(Z/22)?
Are there commutative rings A and B with no homomorphisms between them>
What are the images of Spec(A) and Spec(B) in Spec(AxB)?
Show that R is a product if Spec(R) is disconnected.
Lecture 31
, November 8:
The nilradical is the intersection of the prime ideals. Examples of Spec(R) for some rings R, and discussion of their topology. Algebraic closures exist (read the proof in the book) and are unique (use Zorn's lemma). Definition of left R-module and homomorphism. (Start reading chapter 10.)
Problem session 7
, November 8:
Show that R is a product if Spec(R) is disconnected.
Let t = (1 + sqrt(-7))/2, and show that the ring Z[t]/(t
^{r}
-1) is finite for all r greater than 0.
Produce the ring homomorphism that corresponds to the map f(t)=(t
^{2}
,t
^{3}
) from the affine line to the nodal cubic curve given by the equation x
^{3}
= y
^{2}
in the x-y-plane.
Lecture 32
, November 10:
Modules, homomorphisms, products, coproducts, direct sums, generalized (block) matrices, matrix multiplication. Abelian groups are just Z-modules.
Lecture 33
, November 12:
Free modules, matrices, submodules, quotient modules, the four isomorphism theorems, finitely generated modules. A quotient of a finitely generated module is finitely generated. (If N is a submodule of M we call M an extension of N and M/N.) An extension of finitely generated modules is finitely generated. Noetherian modules.
Homework #10
due Monday, November 29: 7.6: 8, 10, 11; 10.3: 24, 25, 26.
Lecture 34
, November 15:
A quotient of a noetherian module is noetherian. An extension of noetherian modules is noetherian. A direct sum of noetherian modules is noetherian. If R is a noetherian ring, then any finitely generated module is noetherian. Cokernels of homomorphisms. If f is a matrix, and A and B are automorphisms of its source and target, then cokernel f and cokernel BfA are isomorphic modules. Generators and relations for modules: finitely presented modules. If R is a noetherian ring, then any finitely generated module is finitely presented. We will study modules via their presentation matrices, preferably by diagonalizing them.
Lecture 35
, November 17:
Exact sequences, preservation of exactness under isomorphism. Relation between right exact sequences and cokernels. Diagonalization of rectangular matrices. (Different from diagonalization of endomorphisms, and from diagonalization of matrices of bilinear forms.) Normal form for rectangular matrices over a field: we can use elementary row and column operations to get a diagonal matrix containing only 1s and 0s; the only invariant is the number of 1s, also called the rank. Smith normal form for matrices over a Euclidean domain: we can use elementary row and column operations to get a diagonal rectangular matrix where each entry on the diagonal divides the next one.
Lecture 36
, November 19:
Smith normal form for matrices over a PID (using secondary row and column operations). Direct sums of homomorphisms and block diagonal matrices. Kernel, cokernel, image commute with direct sum. Structure theorem for finitely generated modules over a PID.
Problem session 9
, November 20:
What are the subfields of a finite field?
Show the multiplicative group of units in a finite field is cyclic.
Exam 2
, November 22:
Ring theory. The scores were 21, 21, 26, 26, 26, 28, 29, 35, 36, 38, 38, 39, B: 41, 41, 45, 45, 45, 46, 46, 46, 47, 48, 50, 50, 50, 55, 55, 55, 60, A: 65, 65, 71, 71, 71, 73, 75, 75, 75, and 95.
Lecture 37
, November 29:
Proof of the uniqueness part of the structure theorem for finitely generated modules over a PID. (We compute the number of invariant factors divisible by a prime power p
^{k}
as the dimension of a certain vector space over R/pR associated with the module. Review of the proof that the dimension of a finite dimensional vector space is well-defined.) Application: up to isomorphism, how many abelian groups of order 32 are there?
Problem session 10
, November 29:
Up to isomorphism, how many abelian groups of order 30 are there? ... of order 36?
Identify the cokernel of the map f from Z/4 to Z/8 x Z/8 given defined by f(x)=(2x,2x).
Show that if f is an n by n matrix of integers with nonzero determinant, then the cardinality of the cokernel of f is equal to the absolute value of the determinant of f.
Lecture 38
, December 1:
Further applications of the structure theorem to abelian groups; see the second and third problems of the last problem session. Rank, invariant factors, elementary divisors (Chinese Remainder Theorem), reconstructing the invariant factors from the elementary divisors, uniqueness of elementary divisors. Consideration of an F[T]-module M; M is torsion iff the rank of M is zero iff no invariant factors of M are zero iff M is finite dimensional over F.
Homework #11
due Wednesday, December 8: 12.1: 9; 12.2: 10, 16; 12.3: 2, 5, 23.
Lecture 39
, December 3:
Rational canonical form and Jordan canonical form of matrices. Assertion that the characteristic matrix T-A gives a presentation of the module corresponding to the matrix A.
Problem session 11
, December 4:
Write down a matrix over the rationals whose elementary divisors are (T-1)
^{2}
, T+2, T
^{2}
+2.
Find elementary divisors for the elements of GL
_{2}
F
_{2}
.
Find the invariant factors and elementary divisors of the matrix
( 1 2 ) ( ) ( 2 4 )
over the rational numbers.
How many conjugacy classes of elements of order 4 in Gl
_{4}
Q are there? ... in Sl
_{4}
Q? ... in Gl
_{4}
C?
Lecture 40
, December 6:
Proof that the characteristic matrix T-A gives a canonical presentation of the module corresponding to the matrix A. Brief description of the algorithm for computing rational canonical form by using row and column operations to diagonalize T-A. Annihilators of modules and minimal polynomials of matrices. Characteristic polynomial of a matrix.
Lecture 41
, December 8:
An example of two nonsimilar matrices with the same minimal polynomial and the same characteristic polynomial. The Cayley-Hamilton theorem. Finite subgroups of multiplicative groups of fields are cyclic. The exponent of a finite abelian group is analogous to the minimal polynomial of a matrix, and the order of a finite abelian group is analogous to the characteristic polynomial of a matrix. Categories. The examples of categories we've seen. The problem of classifying objects of a category up to isomorphism in various categories. The definition of functor.
Lecture 42
, December 10:
Isomorphism of categories, product of categories, the opposite category of a category, natural transformations, natural isomorphisms, equivalences of categories, free objects, all with examples. We didn't have time to give the categorical definition for colimits and coproducts, limits and products, but the homework from section 7.6 included problems 8 and 10, which cover these notions in particular categories.
Final Exam
, December 14.
The scores were 20, 31, 35, 37, 44, 47, 48, 48, 49, 49, 52, 53, 53, 59, 59, 60, 64, B: 71, 74, 74, 76, 77, 88, 92, 96, 97, 99, 107, A: 119, 119, 121, 123, 124, 131, 136, 142, 149, 155, 155.