MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 14
Due: Friday 15th March: 3:30 PM
Basic Questions
1. Which of the following pairs of numbers are conjugate over Q?
(a) 2 and 6.
(b) 1 + 2 and 1 2.
(c) 4 2 and 2.
2. In Q( 2 + 3), compute 2+
MATH 3031
Abstract Algebra
Assignment 1
Determine whether or not each of the following is a group. In each case
explain why or why not (i.e. if yes, then explain why each of the axioms
is satisfied and if no, then identify which axiom isnt satisfied and e
MATH 3031
Abstract Algebra
Assignment 2
1. In class we defined a subgroup of a group G to be a subset which
is closed under the binary operation and taking inverses. Prove that a
subset which is closed under the binary operation and is itself a group is a
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Midyear Examination
Friday 7th December: 7:00-10:00 PM
Basic Questions
1. Compute the factor group Z3 Z9 / (1, 6) .
The subgroup generated by (1, 6) is cfw_(1, 6), (2, 3), (0, 0), so the factor
group has 2
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 9
Due: Wednesday 28th November: 3:30 PM
Basic Questions
1. Factorise f (x) = x4 + 3x3 + 2x2 + 9x 3:
(a) over Z3 .
Over Z3 , we see that f (0) = 0, f (1) = 0, f (2) = 0, so we see that f (x)
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Midterm Examination
Model Solutions
Basic Questions
1. Which of the following are groups:
(a) The set of functions f : R R such that f (1) = 0 with pointwise
addition (i. e. (f + g )(x) = f (x) + g (x).
Th
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Midterm Examination
Model Solutions
Basic Questions
1. Are the following multiplication tables groups? Justify your answers.
a
(a)
b
c
a
c
a
c
b
a
b
a
c
c
a
c
This is not a group, since it does not have an
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 1
Due: Wednesday 26th September: 3:30 PM
Basic Questions
1. Which of the binary operations in the following table are (a) Commutative
(b) Associative?
a
(i) b
c
d
a
(iii)
b
c
a
a
a
c
c
b
b
b
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 8
Due: Wednesday 21st November: 3:30 PM
Basic Questions
1. Find the remainder of 612345 when divided by 13.
We know that 612 1 (mod 13), so 612345 611 (mod 13). We calculate
62 10 (mod 13),
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Sample Midyear Examination
This sample exam is deliberately longer than the actually midyear. It also
includes only questions from the topics covered after the midterm exam, although the midyear exam will
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 7
Due: Wednesday 14th November: 3:30 PM
Basic Questions
1. Which of the following are rings:
(a) The collection of integers with the usual addition and multiplication
given by a b = ab + a +
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 4
Due: Wednesday 17th October: 3:30 PM
Basic Questions
1. In S4 , let H be the subgroup of permutations that x 4. What is the left
1234
coset of H containing the permutation
?
2413
This is t
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 2
Due: Wednesday 3rd October: 3:30 PM
Basic Questions
1. (a) Show that the collection of symmetries of a regular hexagon is a group
of order 12.
The symmetries of a regular hexagon consist o
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 5
Due: Wednesday 24th October: 3:30 PM
Basic Questions
1. Which of the following functions are homomorphisms?
/ S3 sending to the permutation obtained by restricting
(a) f : S5
to cfw_1, 2,
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 13
Due: Friday 15th February: 3:30 PM
Basic Questions
1. Compute a composition series for S4 .
2. Let G = Z30 , let K = 6 and let H = 3 . Give an explicit description of
/ (G/K)/(H/K).
the i
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 11
Due: Monday 4th February: 3:30 PM
Basic Questions
1. Calculate the dimension of Q[ 5 7] as a vector space over Q.
2. Give a basis of Q[ 1 + 23 i] over Q.
2
3. What is Irr( 3 + 3 3, Q)?
4.
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 12
Due: Monday 11th February: 3:30 PM
Basic Questions
1. Show that it is not possible to trisect an angle of cos1 (0.6). [An angle of
cos1 (0.6) is constructable.]
2. Show that x3 + 2x2 + 4x
MATH 3030, Abstract Algebra
Winter 2013
Toby Kenney
Homework Sheet 16
Due: Wednesday 27th March: 3:30 PM
Basic Questions
1. Let f be an irreducible quartic (degree 4) polynomial over a perfect eld
F . Let K be a splitting eld for f over F . Let the zeros
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 15
Due: Friday 22nd March: 3:30 PM
Basic Questions
1. Find a basis for the splitting eld over Q of x3 4.
2. (a) What is the order of G(Q( 3 2)/Q)?
(b) What is the order of G(Q( 3 2, sqrt3 i)
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 10
Due: Friday 25th January: 3:30 PM
Basic Questions
1. Which of the following are ideals?
(i) The set of all polynomials whose constant term is 0 in Q[x].
(ii) The set of all polynomials a0
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 1
Due: Friday 28th September: 3:30 PM
Basic Questions
1. Which of the binary operations in the following table are (a) Commutative
(b) Associative?
a
(i) b
c
d
a
(iii)
b
c
a
a
a
c
c
b
b
b
c
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 2
Due: Friday 5th October: 3:30 PM
Basic Questions
1. (a) Show that the collection of symmetries of a regular hexagon is a group
of order 12.
(b) Find all subgroups of this group.
2. How man