MATH 124
Homework Assignments
- Assignment #1. Due Wed, Aug 28
- Find two non-zero 2 by 2 matrices mod 5 whose product is the
zero matrix.
- Assignment #2. Due Wed, Sep 4
- p.38, #1-4 and #36.
- Assignment #3. Due Wed, Sep 11
- Assignment #4. Due Mon, Sep 16
- p.64, #27, 28, 36.
- Assignment #5. Due Fri, Sep 20
- p.63, #24, 34. Also, find all subgroups of the integers mod 6
under addition (Hint: There are 4 of them (i.e. 2 proper subgroups)).
- Assignment #6. Due Mon, Sep 30
- p.74, #43, 51, and 45. The first two involve using the
three conditions in Theorem 1.4.14 to show that certain sets are
subgroups. The last problem (#45) is a bit more complicated. You
are asked to prove that a single test (a*b' in H) can be used to
show that H is a subgroup. So, you need to do two parts: Assume
that the three subgroup conditions hold for H and prove that the
shortcut condition holds, and assume that the shortcut condition
holds and prove that the three conditions follow.
- Assignment #6 1/2. Due ???(sometime early October)
- p.85, #26, 27, and 59. I apparently forgot to post
this assignment to the website (hence the funny numbering and uncertainty
about the precise due date).
- Assignment #7. Due Mon, Oct 7
- Assignment #8. Due Fri, Oct 11
- p.101, #1-9.
- Assignment #9. Due Wed, Oct 16
- p.114, #2, 4, 6, 8, 10.
- Assignment #10. Due Wed, Oct 23
- p.116, #26 and p.127 #27, 33.
- Assignment #11. Due Mon, Nov 4
- p.171, #47 and #50.
- Assignment #12. Due Mon, Nov 11
- p.128, #39 and p.178, #24, 30, 33. The statement of
#33 on p.178 is a bit complicated (it involves ideas from
several other problems). Here's an alternate statement which
you can prove instead of #33 (it uses the same proof ideas):
Suppose C is a normal subgroup of G such that a b a^-1 b^-1 is
contained in C for every a, b in G. Show that the factor
group G/C is abelian.
- Assignment #13. Due Fri, Nov 15
- Let R=Z be the integers, and R'=ZxZ be the direct product
ring of the integers with themselves (see example 5.1.7
for direct product of rings). Define a map phi from
R to R' by phi(n) = (n,0) for all n in Z.
- Show that phi is a ring homomorphism.
- Show that phi(1) is a multiplicative identity
for phi[R], but not for all of R'.
- Assignment #14. Due Mon, Nov 18
- p.262, #38, 44 and p.332 #17, 26.
- Assignment #15. Due Mon, Dec 2
- p.331 #4, 25, 27, 34.
- Assignment #16. Due Wed, Dec 4
- 1. Find a generator for the multiplicative group
of the factor ring Z_3[x]/<x^2+1>.
2. Find an irreducible polynomial of degree 3 in
Z_7[x].
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