Homework #4.
Approximate plan for next week: Direct and semi-direct products and
further applications of Sylow theorems (5.4, 5.5).
Problems, to be submitted by Thursday, September 22nd
1. Let G be a
Homework #4.
Approximate plan for next week: Basic applications of Sylow theorems
(4.5, pp.142-146); Recognizing direct and semi-direct products (5.4, 5.5).
Problems, to be submitted by Thursday, Sept
Math 7751, Fall 2009. Solutions to the Final exam.
1. (a) Let R and S be rings with 1. Prove that every ideal of R S has the
form I J where I is an ideal of R and J is an ideal of S .
Note: A common m
Homework #5.
Approximate plan for next three weeks:
Oct 8: Classication of nitely generated abelian groups (5.2)
Oct 13,15: Nilpotent and solvable groups (6.1)
Oct 20,22: Free groups (6.3)
Note: our d
Homework #7.
Plan for next week: Free groups (Section 6.3).
Problems, to be submitted by Thursday, October, 22nd
1. Let n 3 and G = D2n , the dihedral group of order 2n.
(a) Prove that G contains a su
Homework #6.
Plan for next week: Nilpotent and solvable groups (the best approximation
in Dummit and Foote is 6.1, but we will not follow it very closely).
Problems, to be submitted by Thursday, Octob
Homework #9.
Plan for next week: Unique factorization domains (8.3 and 9.3 in DF).
Problems, to be submitted by Thursday, November 12th
1. (a) Let G be a nitely generated group. Use Zorns lemma to sho
Homework #8.
Plan for next week: Finish free groups and presentations by generators
and relations. Start ring theory (perhaps on Tuesday), 7.1-7.4. Sections 7.17.3 contain basic material on ring theor
Homework #10.
Plan for next three classes:
Tuesday, November 17th: Irreducibility criteria in polynomial rings (9.4);
Thursday, November 19th: Finite elds (9.5 + some other stu)
Thursday, November 24t
New Year Homework.
1. Let R be a commutative Noetherian ring and : R R a surjective ring
homomomorphism. Prove that must be an isomorphism. Hint: Consider
the ideals Ker (n ), n N, where n is composed
16. Free groups I
Recall the following elementary fact from Lecture 2: If G is a group and X
is a generating set of G, then any g G can be written as
g = x1 . . . xk where xi X, i = 1, and if
1
k
xi+1
Counting monic irreducible polynomials in Fp [x]
Let p be prime. For n N denote by an the number of monic irreducible
polynomials of degree n in Fp [x]. In this note we shall derive an explicit
formul
24. Irreducibility in polynomials rings
In this lecture all rings are commutative with 1.
Main Problem: Let R be a domain and p(x) R[x] a non-constant polynomial. We want to nd sucient conditions for
26. Completions of rings
26.1. Norms on rings. Let A be a ring and suppose that we have a function
: A R0 , called a norm, such that
(1) x = 0 x = 0
(2) x + y x + y for all x, y A
(3) xy x y for all x
29. Affine algebraic sets, geometric interpretation of HBT
and Nullstellensatz
29.1. Ane algebraic sets. Let k be a eld. For n N let k n be the
n-dimensional ane space over k . We shall think of eleme
27. l-adic integers (continued)
Recall (last time): If A = Z, l 2 an integer and I = (l), the associated
completion AI is denoted by Zl and called the ring of l-adic integers.
Theorem 27.1. The follow
Midterm #1. Due Thursday, October 1st
Directions: Each problem is worth 10 points. The best 4 out of 5 problems
will be counted (but you are encouraged to do all 5). Provide complete
arguments (do not
Homework #3, to be submitted by Thursday, September, 17th
1. (a) Let (G, X, ) be a group action. For a subset S of X we put
GS = cfw_g G : g s = s for any s S (the poinwise stabilizer of S ) and
Gcfw
Homework #2, to be submitted by Thursday, September, 10th
1. (a) Let G be a cyclic group of order n < . Prove that if k Z, then
the mapping : G G dened by (x) = xk is bijective if and only if k is
rel
Homework #1, to be submitted by Thursday, September, 3rd
1. Let A be a set and let f : A A and g : A A be mappings such that
f g = idA (where idA is the identity mapping on A).
(a) Prove that f is sur
Homework #2, to be submitted by Thursday, September, 8th
1. (a) Let G be a cyclic group of order n < . Prove that if k Z, then
the mapping : G G dened by (x) = xk is bijective if and only if k is
rela
Homework #3, to be submitted by Thursday, September, 15th
1. An action of a group G on a set X is called transitive if it has just one
orbit, that is, for any x, y X there exists g G with g.x = y .
(a
Homework #5.
Approximate plan for next three weeks:
Oct 4,6: Nilpotent and solvable groups (6.1)
Oct 13: Free groups (6.3)
Oct 18,20: Free groups and presentations by generators and relations.
Note: o
Homework #1, to be submitted by Thursday, September, 1st
1. Let A be a set and let f : A A and g : A A be mappings such that
f g = idA (where idA is the identity mapping on A).
(a) Prove that f is sur
Math 7751, Fall 2011. Final exam. Due Tuesday, December 6th
Directions: Each problem is worth 15 points. The best 5 out of 6 scores
will be counted with 100% weight, and the lowest score will be count
Homework #6.
Plan for next week: Free groups (6.3)
Problems, to be submitted by Thursday, October, 13th
1. (a) Classify all abelian groups of order 360 = 23 32 5 up to isomorphism.
For each isomorphis
Homework #7.
Plan for next week: Free groups (continued) and presentations of groups
by generators and relators ( 6.3), start ring theory ( 7.1).
Problems, to be submitted by Thursday, October, 20th
1
Counting monic irreducible polynomials in Fp [x]
Let p be prime. For n N denote by an the number of monic irreducible
polynomials of degree n in Fp [x]. In this note we shall derive an explicit
formul
Homework #8.
Plan for next week: Properties of ideals (7.4). Ring of fractions and
Localization (7.5).
Problems, to be submitted by Thursday, October, 27th
1. (a) Let X be a nite set with |X | = n, an
Homework #10.
Plan for next week: Finite elds (9.5 + some other stu), Hilbert basis
theorem (9.6).
Problems, to be submitted by Tuesday, November, 22nd
1. Let R = Z + xQ[x], the subring of Q[x] consis