Modern Algebra I, Fall 2014
Homework 1, due Monday September 15 before class.
Read Chapter 1 of the textbook (Judson, Abstract Algebra).
1. List all subsets of the 3-element set A = cfw_1, 2, 3. How many
subsets does a set with n elements have?
2. Simplif
MODERN ALGEBRA II HOMEWORK 1
DUE WED 10 SEPTEMBER BY 2PM
Problems marked Extra Credit are optional, and should be handed in to me
in class or emailed to neumann@math.columbia.edu with EC in the subject line.
However, the (rare) grade of A+ is for exceptio
MODERN ALGEBRA II HOMEWORK 2
DUE WED 17 SEPTEMBER
Problems labelled Extra Credit should be given directly to Neumann (or by
email to neumann@math.columbia.edu with EC in the subject line) and NOT put
in the homework box.
In class I gave a brief descriptio
MODERN ALGEBRA II HOMEWORK 6
DUE WEDNESDAY OCT 22 2PM
(1) We know that if is a zero of an irreducible polynomial f (x) then the map
: Q[x]/(f (x) Q(), [g(x)] g() is an isomorphism. Applying this
to each of the zeros (see problem (4) of HW5)
1 = 2 + 3, 2
MODERN ALGEBRA II HOMEWORK 3
DUE WED 24 SEPTEMBER
Recall that if d is a squarefree integer (i.e., it is not divisible by any integer
square greater than 1), then the associated quadratic ring is
Z( d) if d 1 mod 4
Z( 1 +
d
2
)
if d 1 mod 4
We denote this
MODERN ALGEBRA II HOMEWORK 4
DUE WED OCTOBER 1
(1) If f = x5 2x3 + 2x2 + 2x 1, g = x3 x2 + 2, nd q and r in Q[x] such
that f = qg + r and deg(r) < deg(g).
(2) Find a generator of the ideal (x5 + x4 + x3 x2 + 2x 2 , x4 1) Q[x].
(3) Find a factorization of
MODERN ALGEBRA II HOMEWORK 5
DUE WEDNESDAY OCT 15
Problems 5 and 6 are extra credit but I suggest reading them anyway, even if
you are not doing extra credit problems. Problem 5 proves a useful result (and is
not a hard problem unless your linear algebra
MODERN ALGEBRA GALOIS THEORY CLASS NOTES AND
HOMEWORK, FALL 2014
These notes will be updated regularly. Any (why?) in the text is a homework
problem that is not to be handed in. Often it is something that you already know
(but may need a reminder) or some
Modern Algebra Fall 2014 Practice Midterm
(1) Mark each of the following true or false (T/F):
Every eld is also an integral domain.
A unit in a ring cannot be a zero divisor.
The set of units in a commutative ring with 1 forms a group under multiplication
Modern Algebra Sample Final Exam
(1) Mark each of the following true or false (T/F):
Every PID is a UFD
Every UFD is a PID
If F is a eld then the polynomial ring F [x1 , . . . , xn ] is a UFD
(unique factorization domain).
The eld of rational functions Q(
Modern Algebra Fall 2014 Practice Midterm
(1) Mark each of the following true or false (T/F):
T
Every eld is also an integral domain.
T
A unit in a ring cannot be a zero divisor.
T
The set of units in a commutative ring with 1 forms a group under multipli
ANSWERS TO MODERN ALGEBRA II MIDTERM 1
(1) Circle the correct answer (T=True or F =False):
F
1. If R is an integral domain then every subring is an integral
domain.
No, for example 2Z is a ring but has no 1.
F
2. Every nite ring is a eld.
T
3. Q[ 3] is th
Modern Algebra Sample Final Exam
(1)
T
F
T
Mark each of the following true or false (T/F):
Every PID is a UFD
Every UFD is a PID
If F is a eld then the polynomial ring F [x1 , . . . , xn ] is a UFD
(unique factorization domain).
F The eld of rational func
MODERN ALGEBRA II MIDTERM 2, NOV 5 2014, ANSWERS
(1) Mark each of the following true or false (T/F):
T
The number is transcendental over Q
F
The number is transcendental over Q( 3 )
Answer: is a zero of x3 3 Q( 3 ).
T
If , C are algebraic over Q then so a
Modern Algebra Fall 2014 Practice Midterm 2
The actual exam will be shorter.
(1) Mark each of the following true or false (T/F):
Every algebraic extension of a eld is a nite extension.
Every nite extension of a eld is an algebraic extension.
An innite eld
Solutions for Midterm exam 2
1. (10 points) Mark the boxes that are followed by correct statements.
The intersection H K of normal subgroups H and K of G is normal
in G.
True. The intersection H K is a subgroup of G, and g(H K)g 1 =
(gHg 1 ) (gKg 1 ) = H
Modern algebra I, fall 2014.
NAME:
Quiz 2
1. Mark the boxes that are followed by correct statements.
The product of two even permutations is even.
Permutation is odd if and only if its inverse 1 is odd.
Dihedral group D5 contains an element of order 3.
Sy
Modern Algebra I, Fall 2014
Homework 2, due Wednesday September 24 before class.
Read Chapters 2, 3.1, 3.2 of the textbook (Judson, Abstract Algebra).
1. Which of the following relations R on sets X are equivalence
relations? Justify your answer.
1) X = Q
Modern Algebra I, Fall 2014
Homework 4, due Wednesday October 8 before class.
Read Sections 4.2, 4.3, 5.1 (the entire section), 6.1, 6.2 (up to and
including Corollary 6.7).
1. (a) Among the following permutations, select those that are even:
id, (1542),
Modern Algebra I, Fall 2014
Homework 3, due Wednesday October 1 before class.
In todays assignment theres no dierence between 2013 and 2014
editions of the textbook.
Read Sections 3.3, 4.1, 5.1 (up to and including Proposition 5.2),
6.1, 6.2 (up to and in
Modern Algebra I, Fall 2014
Homework 6, due Wednesday October 22 before class.
1. We say that an element x G is central if xg = gx for any g G.
The center Z(G) of G is the set of its central elements
Z(G) = cfw_x G : xg = gx g G.
(a) Show that x is centra
Modern Algebra I, Fall 2014
Homework 5, due Wednesday October 15 before class.
Read Sections 5.2 (Dihedral groups), 6.3 (Fermats and Eulers theorems), 9.1 (isomorphism of groups), 9.2 (direct products), 10.1 (Factor groups and normal subgroups), 11.1 (hom
Modern Algebra I, Fall 2014
Homework 7, due Wednesday October 29 before class.
1. Write down a proof that, for a group G and an abelian group H,
the set of all homomorphisms Hom(G, H) from G to H is an abelian
group. (Hint: youll need to explain why the p
Modern Algebra I, Fall 2014
Homework 10, due Wednesday November 26 before class.
1. (10 points) Which of the following groups are simple?
C5 ,
D5 ,
A3 ,
A4 ,
A5 ,
S5 .
2. (10 points) Give examples of subgroups of S5 which have orders
1, 2, 3, 4, 5, 6, 8,
Modern Algebra I, Fall 2014
Homework 9, due Wednesday November 19 before class.
1. Assume that a cyclic group Cp of order a prime number p acts
on a set X. Show that an orbit of Cp either consists of p dierent
points or just one point.
2. The symmetric gr
Modern Algebra I, Fall 2014
Homework 8, due Wednesday November 12 before class.
1. List isomorphism classes of abelian groups of orders (a) 16, (b)
40, (c) 125.
2. Do parts (2),(3), and (6) of Exercise 5 on page 12.6 of Gallaghers
notes (Section 12 on Iso
Name:
Modern algebra I, Midterm exam 1.
1. Mark the boxes that are followed by correct statements.
90 10 (mod 15)
If K is a subgroup of H and H is a subgroup of G, then K
is a subgroup of G.
The symmetric group S5 has order 100.
If g and h commute then g
Name:
Modern algebra I, Midterm exam 2, November 12
1. (10 points) Mark the boxes that are followed by correct statements.
The intersection H K of normal subgroups H and K of
G is normal in G.
Any abelian group of order 8 contains an element of order
4.
T
Modern Algebra I, fall 2014
Practice quiz
Mark the squares that are followed by correct statements.
1.
Composition of two injective maps is injective.
2.
There exists only one set with one element.
3.
The empty set is a subset of any set.
4.
Given any sur
Modern algebra I, fall 2014.
NAME:
Quiz 1
1. Mark the boxes that are followed by correct statements.
(A A) \ A = for any set A.
Composition of bijective maps is bijective.
There exists a surjective map from the set A = cfw_a, b to the
set B = cfw_1, 2, 3.