New Information
Last homework collection 5/2. Last day to hand in extra credit about trancendense of Pi was Wed 4/9
Extra credit opportunity: produce a clear exposition
of a proof of the transcendence of pi. Partial credit for irrationality
of pi. Possible sources: Niven, Bull. AMS 53(1947) p. 509, books
Siegel, Transcendatal Numbers, Niven, Irrational Numbers.
This course is a continuation of Math 504 (which is the prerequisite). Modules and vector spaces will be studied. Fields and Galois Theory will be studied extensively. Additional material on rings and groups will be covered. Chapters 9 - 14, 6 of the text will be covered.
Math 504 Homework Assignments
Homework is collected on Fridays. Problems assigned
through Monday will be collected on Friday. First homework
collection Friday Jan. 17, 2003 (Mon. 1/13 assignment only).
Date Section Problems
Section Problems
12/9 8.3
2,8,9,11 also 1) state ACC, MAX condition and FG condition for rings and ideals.
2) define DCC and MIN condition in general and state for rings and ideals.
3) prove DCC <=> MIN condition (either in general or for rings and
ideals).
12/6 8.2
1-6
12/4 8.1
1c, 2a, 3, 4, 6, 9, 11
12/2 7.4
11,19,26,27,30,32,36,37,38,40 7.5
2, 3, 4, 6
11/22 7.4
2,3a,8,15,20,24
11/18 7.3
34. 35
7.4 1, 4,
5
11/15 7.3
1,2,7,8,11,16,17,19,29-32
11/13 7.2
2, 3c, 4, 6, 7, 9
11/11 7.1
10-12, 14, 15, 17, 24(3), 25, 28
7.2
2
11/8 6.3
1(1st statement), 2, 4
7.1
3, 5-9
11/6 5.5
1, 2, 6, 8, 18
11/4 5.4
10, 11, 13
5.2
16
11/1 5.2
11, 13
10/30 5.2
1ace, 2ace, 3ace, 4-9
10/28 5.1
1,4,5,7,10,11,15,17,18
10/25 4.6
1
10/23 4.5
1,2,5,14,15,17,19,22,25,30,32-36
10/21 4.5
10/18 4.4
1,3,6-12
10/16 4.3
2, 5, 6 1st statement, 11a, 13, 19-24, 26, 27 and
1. Find # conjugates of (1234)(56) in A5 w/o 19-24
2. Find centralizer in S7 of (1234)(567)
3. G nonabelian, p prime, |G|=p^3. Prove: a) |Z(G)|=p, b) if H<G &
|H|=p^2 then Z(G) <= H
10/14 4.2
2, 7 - 9, 11 - 14
10/11 4.1
1-3, 7-9
10/4 3.4
2(Q8), 6, 7 cf. Hungerford for proof of Jordan-Holder Theorem done in class
10/2 3.5
1, 2, 9, 13
9/30 3.3
2, 3, 7
9/27 3.1
3, 4, 11, 14, 15, 20, 22, 24, 36, 37
9/25 3.2
1, 4, 5, 7-12, 16-18
9/23 2.4
1, 3, 14, 16, 17, 19, 20
2.5 1, 4,
7, 10, 11, 12, 13
9/20 2.3
1,3,12,16,23-26
9/18 2.2
1-3, 4Q8, 5a, 9, 10, 11
Prove: if G is a group, H <= G, and |G| = 2|H| then H is normal in G.
If A is a subset of a group G and g is one particular element of G,
does gAg^-1 subset of A imply gAg^-1 =A?
9/16 1.7
4b, 14, 15, 18, 19
prove if f is a homomorphism from G to H and ker f = {e} then f is 1-1
9/11 1.6
1-4, 7, 11, 17, 18, and prove
1) if f is a homomorphism from G to H then |f(g)| | |g| for g in G
2) if f is an isomorphism from G to H then ker f = {e}
9/9 1.5
1
2.1 1bc,2ac,3a,6,8,
10a,15
9/6 1.2
2,3,8 and identify elements in D8 (generators and relations)
and group of symmetries of square described as permutations
9/4 1.3
1, 3(1 only), 5, 7, 13, 14
1.4
1-4
8/30 1.1
1,2,3,5,6,10,11,12,17,20,22,25
1.3 read
8/28 0.3
2, 4, 6, 9-13
1.1 read
8/26 0.1
Prop 1 (2)(3), Prop 2, 1,2,3,5,7
0.2 1bd,
2,3,5,6,9
9/25 1.7.19, 2.2.10, 2.3.1 (draw lattice) 2.3.24,
Prove:
if G is a group, H <= G, and |G| = 2|H| then H is normal in G,
If A is a subset of a group G and g is one particular element of G,
does gAg^-1 subset of A
imply gAg^-1 =A? (prove or disprove)
10/2: 2.4.3, 2.4.17, 2.4.20, 2.5.1, 3.1.3, 3.1.36,
3.2.4, 3.2.16, 3.2.18
remember: "Brevity is the soul of wit." All of
the above problems fit easily on 4 pages.
10/16 4.1: 1-3, 7-9
10/23 4.2.8, 4.2.14, 4.3.5, 4.3.13, 4.3.24, 4.3.27,
4.4.8, 4.4.12
10/30 4.5:14,17,19,25,32,35; 4.6:1
11/6 5.15, 18; 5.2: (1,2,3,4)c, 5, 8, 11
11/13: 5.2.16, 5.4.13, 5.5.8, 5.5.18, 6.3.1(1st), 7.1.3
11/22: all assigned 11/11, 11/13, 11/15
12/4: 7.3:34, 35; 7.4: 2,3a,4, 5, 8,15,20,24
Problems to be collected:
Math 504 Extra Credit Opportunities
Aubrey's Lattice Challenge (extra credit) updated 10/4
Euclidean Domain: Prove Dummit and Foote's definition of a Euclidean domain (norm with division algorithm) implies the standard one, i.e. there is a norm N that satisfies the division algorithm and for which N(a) <= N(ab) if b not 0.
Proof and Example : A finitely generated nontrivial group has a maximal proper subgroup. Example that shows A nontrivial group need not have a maximal subgroup.
This is the first semester of a 2 semester sequence of graduate courses in abstract algebra. Two semesters of undergraduate abstract algebra and one semester of undergraduate linear algebra are prerequisites. Groups will be studied extensively. Rings will be studied. Chapters 1 - 5, 7, 8 of the text will be covered as will parts of Ch. 6 and 9.policies
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