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 nite group, P a Sylow p-subgroup of G for some p and H
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, September 24th
1. Let G be a group such that G/Z (G) is cyc
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 mistake was the wrong assumption that every subring of
R
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 discussion of niplotent groups and free groups will be s
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 subgroup isomorphic to D2k for any k | n.
(b) Prove that
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, October, 15th
1. (a) Classify all abelian groups of order 36
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 show that G
has a maximal subgroup (recall that a maximal
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 theory covered in undergraduate courses
and will only be bri
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 24th: Local rings (no specic section in DF)
1. The followi
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 with itself n times.
2. Let R be a commutative Noether
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 = xi for some i, then i+1 = i (we allow k = 0).
In thi
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
formula for an and in particular prove that an > 0 for any n
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 p(x) to be irreducible in
R[x].
We are mostly intereste
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, y A
Dene the metric d on A associated with the norm
b
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 elements of k n as
points, not vectors. Any f k [x1 , . . .
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 following hold:
(a) If p is a prime, then Zp is a domain and
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 skip steps). State clearly any result you are referrin
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_S = cfw_g G : g S = S (the stablizer of S ).
Prove t
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
relatively prime to n.
(b) Let G be an arbitrary nite grou
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 surjective and g is injective
(b) Show by example that f n
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
relatively prime to n.
(b) Let G be an arbitrary nite group
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) Let (G, X, .) be a group action. Prove that if x, y X
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: our discussion of niplotent groups and free groups will
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 surjective and g is injective
(b) Show by example that f n
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 counted with
1
33 3 % weight, so the maximal possible total
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 isomorphism type, state the corresponding elementary divisors for
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. Let n 3 and G = D2n , the dihedral group of order 2n.
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
formula for an and in particular prove that an > 0 for any n
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, and let F = F (X ) be the
(standard) free group on X . Pr
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] consisting of polynomials whose
constant term is an integer.