MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 14
Model Solutions
Basic Questions
1. Which of the following pairs of numbers are conjugate over Q?
(a) 2 and 6.
These are not conjugate, since Irr( 2.Q) = x2 2, while Irr( 6.Q) =
x2 6.
(b)
MATH 3030, Abstract Algebra
Winter 2012
Toby Kenney
Sample Midterm Examination
This practice exam deliberately has more questions than the real
midterm. Some of the theoretical questions are directly from the
notes, and some are new, requiring a little th
MATH 3030, Abstract Algebra
Winter 2012
Toby Kenney
Midterm Examination
Monday 18th February: 2:35-3:25 PM
Basic Questions
1. Let R = Z4 Z2 . Let I be the ideal of R generated by (2, 1).
(a) What is the ideal I ?
R is commutative, so we only need to consi
MATH 3030, Abstract Algebra
Winter 2012
Toby Kenney
Sample Final Examination
This practice exam deliberately has more questions than the real
exam. Some of the theoretical questions are directly from the
notes, and some are new, requiring a little thought
MATH 3030, Abstract Algebra
Winter 2012
Toby Kenney
Sample Midterm Examination
Model Solutions
Basic Questions
1. Give an example of a prime ideal which is not maximal.
In the ring Z Z, the ideal cfw_(0, a)|a Z is prime but not maximal.
2. Let R = M2 (Z2
MATH 3030, Abstract Algebra
Winter 2012
Toby Kenney
Sample Final Examination
This practice exam deliberately has more questions than the real
exam. Some of the theoretical questions are directly from the
notes, and some are new, requiring a little thought
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 15
Model Solutions
Basic Questions
1. Find a basis for the splitting eld over Q of x3 4.
3
3
The splitting eld is Q 3 4, 23 i . One basis for this eld is cfw_1, 3 4, 2 3 2, 23 i, 32 4 i, 3
MATH 3030, Abstract Algebra
Winter 2013
Toby Kenney
Homework Sheet 17
Model Solutions
Basic Questions
1. (a) Is the regular 120-gon constructable?
120 = 23 3 5. 3 and 5 are Fermat primes, so the regular 120-gon is
constructable.
(b) Is the regular 28-gon
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 11
Model Solutions
Basic Questions
1. Calculate the dimension of Q[ 5 7] as a vector space over Q.
5
5
5
A basis for this vector space is cfw_1, 5 7, 72 , 73 , 74 , so the dimension
is 5.
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 13
Model Solutions
Basic Questions
1. Compute a composition series for S4 .
We know that A4 is a normal subgroup of S4 , so we can choose A4 as one element in the composition series. Next we
MATH 3030, Abstract Algebra
Winter 2013
Toby Kenney
Homework Sheet 16
Model Solutions
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 of f in K be ,
, a
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 10
Model Solutions
Basic Questions
1. Which of the following are ideals?
(i) The set of all polynomials whose constant term is 0 in Q[x].
Let f and g be polynomials in Q[x] with constant ter
MATH 3030, Abstract Algebra
FALL 2012
Toby Kenney
Homework Sheet 12
Model Solutions
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.]
Trisecting an angle of cos1 (0.6) means const
MATH 3030, Abstract Algebra
Winter 2012
Toby Kenney
Midterm Examination
Monday 18th February: 2:35-3:25 PM
Basic Questions
1. Let R = Z4 Z2 . Let I be the ideal of R generated by (2, 1).
(a) What is the ideal I ?
(b) What is the factor ring R/I ?
2. What