Math 412-Summer 2012

Final Course Grades Math 412

I submitted these and they should appear on your transcript soon. Here is the grade distribution. Enjoy the remainder of your summer.

Special Note

If you took Math 411 Summer 2012, that website is appended below. It will contain information on the last homework, the final exam and the distribution of course grades (by Monday July 23).

412 Syllabus

Here is the 412 syllabus that explains everything about how the class will be organized.

412 Homework

All homework is due by 2pm on the indicated day. No late homework accepted.

412 Homework 1: Due July 27. solutions
1. 3.7: 1,2,3,4,8,10,12,
2. 4.1: 2,4,10,12.
3. Let G= Rationals-{2} and define a*b=ab-2a-2b+6. Show that with this operation G becomes a group. What is the identity element? what is the formula for the inverse of a? What is the inverse of -1? What is the inverse of 3/2?
4. 4.2: 4,8.
5. 4.4: 2,3.
6. grading: 1. 3.7: carefully grade #4,8,12 for 10 pts each. 5pts each for the other 4 (#1,2,3,10) if they did them. that gives 50 pts.
  2. 4.1: carefully grade #2,12 for 10 pts each and 5pts for each of #4,10. that gives 30pts.
  3. for the funny operation on Q-{2}, carefully grade worth 30 pts.
  4. 4.2: carefully grade both #4,8 for 10 pts each. that gives 20 pts.
  5. 4.4: carefully grade #3 for 10 pts and 5pts if did #2.
  grand total= 50+30+30+20+15=145pts
7. stats:high=135,low=49,median=113.
412 Homework 2: Due August 3. solutions
1. Describe the containment lattice of ALL of the subgroups of D_4. (For this question, you may assume that the order of any subgroup of D_4 is a divisor of 8, which we will prove later.) You need to justify your answer clearly.
2. Describe the containment lattice of ALL of the subgroups of Z_30. You need to justify your answer clearly.
3. 4.4: 10,12,13,24,25
4. 4.6: 2,3,14
5. Define a 2x2 matrix by
```
         /  cos(t)       sin(t) \
A(t) =   |                      |
         \  -sin(t)     cos(t)  /
```
  As discussed in class, this will rotate vectors in R^2 by t radians clockwise.
  1. Show that A(t) is in GL_2(R).
  2. Find a matrix H in GL_2(R) that reflects vectors in R^2 across the x-axis.
  3. Using the above, explicitly write out the 2x2 matices in GL_2R that form the groups D_3, D_4 and D_6. In this way, we have constructed finite subgroups of GL_2(R) that are these dihedral groups.
6. . Do the task at the bottom of page 239. Lets all agree to use the same numbering of the four diagonals. Label the 8 vertices in the picture figure 1 like this:
```
     a --- b
   / |     /|
  c ---  d  |
  |  |   |  |
  |  e ---  f
  |/     |/
  g ---  h


1= diagonal ah
2= diagonal ed
3= diagonal cf
4= diagonal gb
```
7. 4.7: (1,2,6,7,8,9,10,11,12,13). Note: On #10, omit the cayley table for A_4 .
8. grading:1. carefully grade 20pts
  2. carefully grade 20 pts
  3. carefully grade 12, 24 for 10 pts each; 5 pts if did each of the other three. total=35pts
  4. carefully grade each for 10 pts, total =30pts
  5. carefully grade for 30pts
  6. 10pts if tried, but do not attempt to grade.
  7. carefully grade 2,9,11,13 for 10pts each and 5pts for 1,6,7,8,12 if tried. omit 10. total=65pts
  grand total=210pts
9. stats:high=195, low=90, median=163.
practice midterm and actual midterm and distribution of scores:
412 Homework 3: Due August 10.solutions
1. 5.1: 2,3,10,12,14,18,19
2. 5.2:2,3,8,13,18.
3. - In A_4, describe the left cosets of K consisting of the identity and the 2-2 type permutations.
  - You will show K is normal in one of the exercises below. Write out the Cayley table of A_4/K. What familiar group is this isomorphic to?
4. 5.3: 5,6(i,ii),7(i,ii), 8.
5. 6.1: 1,2,3,4,6.
6. Determine if U_{21} is solvable and if so, construct the required chain.
7. Is U_{11} a simple group?
8. grading: 1. 5.1: carefully grade 2,12,18,19 for 10 pts each and 5pts each if did 3,10,14. total=40+15=55pts.
  2. 5.2: carefully grade 2,18 for 10pts each and 5pts each if did 3,8,13. total=35pts
  3. carefully grade for 20pts
  4. 5.3: carefully grade 6i,ii and 7i,ii for 10 pts each and 5pts if did 5, 8. total= 20+10=30pts
  5. 6.1: carefully grade 2,4,6 for 10pts each and 5pts if did 1,3. total =30+10=40pts
  6. carefully grade for 20pts
  7. carefully grade fo 20pts
  
  grand total=55+35+20+30+40+20+20=220pts.
9. stats:high=205,low=0,median=145
practice final
412 Homework 4: Due August 17.
1. Find the multiplicative inverse of y=2+2^(1/3)+2^(2/3) in Q(2^(1/3) by using the constructive proof that this is a field.
2. 6.2: 1i,3,4.
3. 6.5:1ii,3,4.

412 Office Hours


who                   where      when
=============================================================
Dave Collingwood     C545PDL   

July 27, 11am-1pm
August 3, 730-830 in our classroom
August 10, 11am-1pm.
August 17, 730-830 in our classroom.


-------------------------------------------------------------
Allison Beckwith     C-8H,PDL     

July 26, 1-2pm
Aug 2, 1-3pm
Aug 9, 1-2pm
Aug16, 1-3pm
=============================================================

Math 411-Summer 2012

Syllabus

Here is the syllabus that explains everything about how the class will be organized.

Homework

All homework is due by 2pm on the indicated day. No late homework accepted.

Homework 1: Due June 20. McNuggets and McNuggets solution worth 20 points.
stats: Max Possible=20,High=20, Low=0, Median=20.
Homework 2: Due June 25. Solutions
stats: Max Possible=135, High=130, Low=69, Median=115.
1. 1.2: 2,6,7,8
2. 1.3: 1,2,4,7,10
3. 1.4: 2,3,4
4. Suppose you can only buy McNuggets in boxes of sizes 4 and 11. Show there is a largest total number of McNuggets you cannot buy; i.e. there is a largest (4,11)-inaccessible integer. Justify your answer following the approach used in homework 1.
5. Go online to "www.wolframalpha.com".
  - Enter GCD(2468,8642) to get the gcd of 2468 and 8642 and turn in your answer.
  - Enter FACTOR(8642) to find prime factorization of 8642 and turn in your answer.
  - Enter DIVISORS(8642) to find divisors of 8642 and turn in your answer.
  - Enter "primes<=100" to get a list of primes under 100 and turn in your answer.
  - Enter "1000<=primes<=1020" to get a list of primes between 1000 and 1020 and turn in your answer.
  - Enter "Solve(x^2 +2x-5=0)" to solve the quadratic and turn in your answer.
  - Enter "Solve(x^3 + x^2 +x +1=0)" to solve the cubic and turn in your answer.
  - Enter "Solve(x^3-x^2-x-1=0) to solve the cubic and turn in your answer.
  - Enter "Solve(x^4-1=0) to solve the quartic and turn in your answer.
  - Enter "Solve(x^5-6x+3=0) to solve the fifth degree equation and turn in your answer. What is different about the output of this answer and the previous four examples?
6. Grading: 1.2: carefully grade #8 for 10pts; effort credit for #2,6,7 at 5pts each.total of 25pts
  1.3: effort credit of 5pts each for #1,2,7,10i,10iii. carefully grade #4 for 10 points and #10ii for 10pts.total of 45pts
  1.4: carefully grade #2 and #4 for 10 pts each and 5pts for turning in #3 for effort. total of 25pts.
  (4,11)-mcnugget problem, 20 pts carefully grade. use same template as homework 1.
  WolframAlpha questions: 2 pt each for doing each part for a total of 20pts.
  Grand total homework #2=25+45+25+20+20=135pts.
Homework 3: Due July 6. Solutions
stats: Max Possible=190, High=186, Low=27, Median=158.
1. Let a,b,c be three non-zero integers.
  - Define gcd(a,b,c)
  - Formulate the analog of Thm 1 p16 for gcd(a,b,c).
  - Explain why gcd(a,b,c)=gcd(gcd(a,b),c)=gcd(a,gcd(b,c)).
2. Find gcd(36,90,126) and write it as an integer linear combination of a, b and c.
3. Find gcd(6,9,20) and write as a integer linear combination of 6, 9 and 20. These were the key McNugget problem numbers.
4. Find integers a,b so that
  
  (29/210) = (a/10) + (b/21)
5. 1.7: 1,2,3,6,7,12.
6. Find the smallest non-negative remainder after 3^(99) is divided by 5.
7. Find the smallest non-negative remainder after 2^(510) is divided by 511.
8. Find the smallest non-negative remainder after 30^(30) is divided by 7.
9. 2.1: 1,2,3,5,8,11,14,15
10. 2.4:1,3,6,7,8,9,10.
11. Go to
  http://mathworld.wolfram.com/McNuggetNumber.html
  http://mathworld.wolfram.com/FrobeniusNumber.html
  and read about McNugget numbers and Frobenius numbers.
Grading: 1. 5pts each part, total 15.
2. 5pts if did.
3. 5pts if did.
4. 10 pts carefully grade.
5. 5pts each for 1,2 3,6,7 if did. 15 pts carefully grade 12. total 40pts
6. 5pts if did.
7. 5pts if did.
8. 10 pts carefully grade.
9. 5pts each if did 1,2,3,5,11,15. 10pts each for 8 and 14 carefully grade. total 50 pts.
10. 5pts each if did 1,3,7,9,10. 10pts each for 6 and 8 carefully grade. total 45 pts.
11. do not grade
Grand total :15+5+5+10+40+5+5+10+50+45=190pts.
Sample Midterm (July 2) and solutions . I guess part 5 did not get posted. here is the argument. First, in Z_p, all nonzero elements are invertible mod p, since all nonzero elements are relatively prime to p (see friday lecture notes). for part (a), we prove the contrapostive: If [x] not [0] and [y] not 0, then [x][y] not 0. To prove it, suppose [x][y]=0. Then mult both sides by [x^-1] and [y^-1] (the inverses mod p). Now we have [x^-1][y^-1][x][y]=[0]. But the left side is [1], a contradiction. To prove (b), rewrite as [x]^2-[1]=[0], factor as difference of squares: ([x]-[1])([x]+1)=[0]. By (a), one of the two factors is zero. If [x]-[1]=[0], then [x]=[1]. If [x]+[1]=[0], then [x]=-[1]=[-1]=[-1+p]=[p-1]. These are the only solutions.
Actual Midterm
stats:Possible=75,High=75,Low=37,Median=70.
Homework 4: Due July 13. Solutions
1. 2.5:4,7,8,10,13,14,15..
2. 2.6:1,10.
3. 3.1:1,2,3i,5i,5iii,5v,7,10,12.
4. 3.2:1,3,4i,5-8.
5. Using the ideas laid out in class on July 9, determine if Z_7[i]={a+bi | a,b in Z_7 and i^2=-1} is a field.
6. grading: 2.5: carefully grade 7,10,14 for 10pts each; 5pts for doing each of the other 4. total 50pts 2.6: do not carefully grade, 5 pts each if did, total 10 pts. 3.1: carefully grade 2,5i,10 for 10pts each; 5pts for doing each of the other 6. total 60pts. 3.2: carefully grade 1,4i,8 for 10pts each; 5pts each for each of the other 4. total 50pts last problem:carefully grade for 10pts. total possible=50+10+60+50+10=180pts.
Homework 5: Due July 18. Solutions
1. 3.4: 4i,6,7.
2. 3.5:4i
3. This question investigates the irreduciblity of p(x)=x^5+x^4 + 2x^3 + 2x +2 in Q[x].
  1. List the degree 2 polynomials in Z_3[x]. Circle the irreducible ones and explain why they are irreducbile.
  2. Show [p]_3 is irreducible in Z_3[x], concluding p is irreducible in Q[x].
4. grading: 3.4: carefully grade 4i, 7 for 10 pts each and 5 pts if did 6. total=25pts
  3.5: 5 pts if did 4i, do not carefully grade. total=5pts
  last question: carefully grade. first part worth 20 pts. second part worth 20 pts. total = 40 pts
  grand total homework #5=25+5+40=70pts.
Sample Final (July 16) and solutions
Actual Final stats: high=98, low=57,median=82.
Course Grade Distribution:

Office Hours


who                   where      when
=============================================================
Dave Collingwood     C545PDL   

June 22, 11am-1pm
June 29, 11am-1pm
July 6, 730-830am  in our classroom if sieg is open, my office otherwise 
July 11, 11am-1pm.
July 16, 11am-1pm.
July 18, 730-830am in our classroom.

-------------------------------------------------------------
Allison Beckwith     C-8H,PDL     

June 19, 1-2pm 
June 25, 1230-130pm 
July 5,1-3p.
July 12, 1-2pm
July 17, 1-3pm.
=============================================================