HOMEWORK #9, DUE THURSDAY APRIL 25TH
1. Herstein, Chapter 4, 4, 2: Let R be the Gaussian integers and
let M be the subset of Gaussian integers a + bi such that a and b are
divisible by 3. Show that M is an ideal and the quotient R/M is a eld
with 9 elemen
1. (15pts) (i) Give the denition of a normal subgroup.
(ii) Give the denition of a group homomorphism.
(iii) Give the denition of An , the alternating group.
2. (15pts) (i) Exhibit a proper normal subgroup V of A4 . To which
group is V isomorphic to?
SECOND PRACTICE MIDTERM
You have 80 minutes. This test is closed book, closed notes, no calculators.
There are 6 problems, and the total number of
points is 100. Show all your work. Please make
your work as clear and easy to follow as possible.
HOMEWORK #8, DUE THURSDAY APRIL 18TH
1. Herstein, Chapter 4, 3, 1.
2. Herstein, Chapter 4, 3, 2.
3. Let I and J be ideals of a ring R.
(i) Show that I J is an ideal of R.
(ii) Show that
I + J = cfw_ i + j | i I, j J
is an ideal of R.
(iii) Let IJ be the
1. (30pts) Give the denition of a group.
(ii) Give the denition of an automorphism of groups.
(iii) Give the denition of Dn , the dihedral group.
(iv) Give the denition of an ideal.
(v) Give the denition of a principal ideal domain.
(vi) Give the deniti