Text:  Thomas Hungerford "Algebra", ISBN 0-387-90518-9 or 3-540-90518-9
 

Office Hours: By appointment. Other good times to talk are right before or right after the class. It is also possible to ask questions by email, which will be generally answered within 24 hours, often sooner. The more detailed your email question is, the more detailed the reply message will be. My email is borisov@rutgers.edu.
 

Final/midterm Examinations: There will not be any.
 

Homework: There will be an extensive homework requirement. An average of five homework problems will be assigned after each class. I will collect a week worth of homework each Thursday, starting from January 29. Collaboration on homework assignemnts is allowed and strongly encouraged, but each person must write up their solutions. Homework can be handwritten or typed up. If you can not make it to class when homework is due, you can take photos of your homework and email me at borisov@math.rutgers.edu. You can also submit homework by email if that is your preference.
 

Weather/Covid rules: If in-person classes are canceled, we will move online to our Canvas site.
 

Schedule of Lectures (updated periodically). Please try to read the relevant textbook sections before the lecture.

January 22 Section 5.1
Homework: page 241; 6 (hint: use the fact that the polynomial ring K[x_1,...,x_n] is a UFD), 8, 14, 17, 18.

January 26 Section 5.2
Homework: page 255; 2, 5, 7, 11, 15.

January 29 homework from the last two lectures is due
Section 5.3
Homework: page 254; 3(a), 5, 8, 20, 23.

February 2 Sections 5.4, 5.5
Homework: page 277; 6, 12; page 281; 3, 6, 8.

February 5 homework from the last two lectures is due
Sections 5.7, 5.8
Homework: page 296; 4, 8; page 300; 8, 9;
Problem: Calculate the n=10 cyclotomic polynomial over Q.

February 9 Sections 5.9, 6.1
Homework: page 309; 1, 3, 4;
Problem: Suppose that an irreducible cubic polynomial over Q has three real roots. Prove that it is impossible to have a formula for these roots involving repeated radicals, starting with rational numbers so that all intermediate radicals are real numbers.
Problem: Suppose that a complex number x is an algebraic integer (over Z) and is such that all of its Galois conjugates have length 1. Prove that x is a root of unity.

February 12 homework from the last two lectures is due
Sections 8.1; Localization.
Homework: page 377; 1, 2, 3, 7;
Problem: Let A be a commutative ring and let S be a multiplicative system in A. Prove that localization by S sends short exact sequences of A-modules to short exact sequences of A_S modules.

February 16 Sections 8.2-8.3
Homework: page 382; 13; page 386; 9, 12, 15, 16.

February 19 homework from the last two lectures is due
Sections 8.3 - continued
Homework: page 382; 14, 15; page 386; 8, 13, 14.

February 23 Sections 8.4
Homework: Problem (very simple). Let A be a local ring with maximum ideal m. Prove that if A is Noetherian, then m/m^2 has finite dimension over A/m. page 393: 3 (this uses Nakayama's Lemma), 6(a), 8, 9.

February 26 homework from the last two lectures is due
Sections 8.5
Homework: page 400; 2, 3, 4, 8, 9.

March 2 Sections 8.7
Homework: page 413; 1, 4, 6, 7, 8.

March 5 homework from the last two lectures is due
Sections 8.6
Homework: page 407; 1, 2, 9, 10.

March 9
Sections 9.1
Homework: page 423; 1(a,b), 2, 5, 6.

March 12 homework from the last two lectures is due
Sections 9.1-contuinued. Quaternions.
Homework: page 423; 3.
Problem: Consider the linear map over R that sends the quaternion a + b i + c i + d k to a - b i - c i - d k. Verify that this is an anti-isomorphism of the ring of quaternions.
Problem: Let H_1 be the subset of quaternions of length 1. Verify that it forms a group under multiplication. Verify that this group is naturally isomorphic to SU(2) (2 x 2 complex unitary matrices of determinant 1).
Problem: Let H_1 be as above. Prove that it acts on the space R^3 which is the span of i, j, k by conjugation. Prove that this action gives a natural homomorphism of SU(2) to SO(3,R), with kernel \pm 1. (Remark: this is not needed for the proof, but SU(2) is simply-connected, because it is a three-sphere, and this is the universal cover of SO(3,R)).

March 23 Sections 9.2
Homework: page 432; 7, 8, 9, 11.

March 26 homework from the last two lectures is due
Sections 9.3
Homework: page 442; 4, 11, 12.
Problem: Let Q be the field of rational numbers and let Q[C_n] be the group ring of the cyclic group of order n. Prove that this group ring is isomorphic to the direct product of the cyclotomic fields Q(d-th roots of 1) over all positive divisors d of n.

March 30 Sections 4.3-start; Clifford algebras
Homework: page 198; 5.
Problem: Let H be the group of order 2^(n+1) defined in class, generated by h_1,...,h_n and c, with relations h_i^2 = c, c^2 = id, h_i h_j = c h_j h_i, c h_i = h_i c. Let H_1 be the subgroup of index two of H defined by the property that the number of occurrences of h_i is even. Prove that the center of H_1 has size 2 if n is even and size 4 if n is odd (this implies the statement about the structure of the even Clifford algebra that I made in class.)
Problem: Let V be a complex vector space of dimension n and let W = Sum_k \Wedge^k V be its exterior algebra. Consider the action on W of elements of V via wegdge product x -> w \wedge x, and the action of contractions by elements of V^* on W. Prove that together they generate the action of the Clifford algebra CL_(2n)(C).
Problem: Let f:A -> B be a surjective morphism of left R-modules. Prove that if there exists a morphism of left modules g: B->A such that fg = id_B, then g is injective and there exists a left submodule C of A such that A is the direct sum of g(B) and C.

April 2 homework from the last two lectures is due
Sections 4.3
Homework: page 198; 6, 7, 8, 9.

April 6 Sections 4.4-4.5
Homework: (all rings are commutative)
Problem: Give an example of a ring R and a short exact sequence of R-modules 0->A->B->C->0, and a module D such that both Hom(D,_) and Hom(_,D) applied to this sequence break the right-exactness.
Problem: Prove that if A is a finitely generated abelian group then (A tensored with A) = 0 implies A = 0. Here tensoring is done over the ring of integers.
Problem: Prove that if D is a free R-module, then tensoring with D sends short exact sequences to short exact sequences. Use it to prove that tensoring with a projective module sends s.e.s. to s.e.s.

April 9 homework from the last two lectures is due
Sections N/A
Homework:
Problem: Compute Tor_i(Z/mZ,Z/nZ) for all nonnegative i. Here m and n are positive integers, and the ring is Z.
Problem: Let R=K[x,y] be the polynomial ring in two variables over a field K. Consider the complex of R-modules
0-> R -> R + R -> R -> R/(x,y) -> 0 where the maps are given by
f -> (x f, y f), (f, g) -> y f - y g, f -> f mod(x,y)
respectively. Prove that this is a long exact sequence, which gives us a projective resolution of R/(x,y).
Problem: Let ...-> C_3 -> C_2 -> C_1 -> ... be a complex of R-modules with the differentials denoted by d. Suppose that there are maps s_i : C_i -> C_{i+1} such that
ds + sd = id. Prove that this complex has zero homology.

April 13 Hilbert functions of graded rings and modules. Sections N/A
Homework:
Problem: Let a_n, n in Z be a (two-sided) sequence of complex numbers that satisfies the following properties.
(1) There exists n_0 such that a_n = 0 for all n < n_0.
(2) There exists a positive integer N and N polynomials p_i(n), such that a_i(n) = p_i(n) for all large enough n with n = i mod N.
Prove that the generating function Sum_n a_n t^n is a Laurent polynomial of a rational function f(t)/g(t), f(t) is a Laurent polynomial in t, and all roots of g are N-th roots of 1.
Hint: run induction on the degrees of p_i.

Problem: Prove the converse of problem 1, i.e. prove that the coefficients of the Laurent series of f(t)/g(t) satisfy (1) and (2). Hint: first consider the Laurent series of 1/(1-t^N)^k.

Problem: Let A be a graded ring and let I be a homogeneous ideal in A. Prove that I is a prime ideal iff for any two homogeneous elements x y with xy in I, at least one of x,y lies in I.

April 16 Hilbert Syzygy Theorem. homework from the last two lectures is due
Sections N/A
Homework:
Problem: Let J be the ideal in the polynomial ring A = K[x1,x2,x3,x4] which is the kernel of the ring homomorphism g: A -> K[y1,y2] that sends x1 -> y1^3, x2 -> y1^2 y2, x3 -> y1 y2^2, x4 -> y2^3.
(a) Describe the image of the homomorphism g.
(b) Compute the Hilbert series of A/J and J.
(c) Prove that J can not be generated by two homogeneous elements.

Problem: Use Hilbert Syzygy Theorem to give an alternative proof of the fact that Hilbert function of a finitely generated graded module over a polynomial ring K[y_1,...,y_n] with all degrees of y_i equal to one is given by f(t)/(1-t)^n where f(t) is a Laurent polynomial.

Problem: Let A = K[x1,x2,x3] be the graded polynomial ring with deg x_i = 1. Suppose that f(x1,x2,x3) and g(x1,x2,x3) are homogeneous polynomials of degrees m and n respectively, without common factors.
(a) Find the Hilbert function of the quotient ring A/(f,g).
(b) Prove that the dimension of the N-th graded piece of A/(f,g) is equal to m n for N large enough.

April 20 Koszul complex continued. Balancing of Tor.


April 23 Rings of invariants. Molien Series. homework from the last two lectures is due
Sections


April 27 Groebner bases.


April 30 Groebner bases continued.


May 4 Hensel Lemma.