Math 370 Algebra Homework

Unless otherwise noted, exercises are from the course text book, Artin's Algebra.

Note that M stands for Artin's "Miscellaneous" exercises section.
Note that *starred problems are optional/hard/extra credit.

• Due Friday Dec 8th 4 pm:
  

  Final exam - have fun!



• HW 8 Due Wednesday Nov 22th (4 pm sharp!):
  

  chapter	page	exercises
  7	262	1.1
  7	263	1.4, 1.5, 1.6, 2.9, 2.10
  7	264	2.17
  4/7		*Additional problem.

• HW 7 Due Tuesday Nov 14th:
  

  chapter	page	exercises
  4	145	2.2
  4	146	3.1, 3.7
  4	148	4.13
  4	150	7.2
  4	152	8.7, 8.13, 8.19
  4	154	*M.17, M.18

• Due Thursday Nov 2nd:
  

  Midterm exam - have fun!



• HW 6 Due Thursday Oct 19th:
  

  chapter	page	exercises
  2	76	10.10
  3	106	3.6, 3.7, 3.10, 3.14, 4.3, 4.5
  3		The following problems have to do with finite dimensional vector spaces over finite fields. Let V be an Fp-vector space of (finite) dimension n. a) How many vectors does V have? b) How many k-tuples of linearly independant vectors does V have? c) How many n x n matrices of rank k are there over Fp? In particular, what's the order of GLn(Fp)? d) What is the isomorphism type of the underlying (finite) abelian group (V,+)? e) Show that the automorphism group Aut((V,+)) of the abelian group (V,+) is isomorphic to the group GL(V) of Fp-linear vector space isomorphisms V -> V and that this group is also isomorphic to GLn(Fp). f) Compute the order of Aut(Z/2Z x Z/2Z x Z/2Z). Can you write down an automorphism of order 7?

• HW 5 Due Tuesday Oct 10th:
  

  chapter	page	exercises
  2		Find all subgroups of S4 and describe their isomorphism types. Which ones are conjugate? Which ones are normal? For the normal subgroups find the corresponding quotient groups. Which isomorphism classes of groups arise as subgroups (resp. as quotients) of S4?
  2		Prove: Z/nZ x Z/mZ is isomorphic to Z/nmZ if and only if gcd(m,n)=1.
  2		Extra credit: Calculate the order of GL2(Fp), and identify the isomorphism type of GL2(F2).
  3	104	2.1, 2.10, 2.7, 2.8, 2.11
  3	105	2.17

  Solutions now available for selected problems from this assignment.

• HW 4 Due Tuesday Oct 3rd:
  

  chapter	page	exercises
  2	70	2.10, 2.11, 2.16
  2	71	2.20, 3.11, 3.14 (also for Z/8Z) Also prove or find a counterexample: If G is cyclic then Aut(G) is cyclic? If G is abelian then Aut(G) is abelian?
  2	72	4.8, 4.9, 4.11
  2	74	6.8

  Solutions now available for selected problems from this assignment.

• HW 3 Due Tuesday Sept 26th:
  

  chapter	page	exercises
  2	71	3.4, 3.5, 3.6, 3.12
  2	72	3.15, 4.4, 4.17
  2	73	4.22, 4.23

  Solutions now available for selected problems from this assignment.

• HW 2 Due Tuesday Sept 19th:
  

  chapter	page	exercises
  2	70	2.1, 2.2, 2.3
  2	71	3.7
  2	72	3.16, 4.16

  Solutions now available for selected problems from this assignment.

• HW 1 Due Tuesday Sept 12th:
  

  chapter	exercises
  1	3.12, 3.13, M.6, M.7, *M.8

  Solutions now available for selected problems from this assignment.