quiz 5 algebraic structures
Name: DrSarah Greenwald
Start Time: Aug 13, 2000 19:13 Time Allowed: 20 min
Number of Questions: 6

Question 1  (2 points)

Theta_n = {cos(2*Pi*k/n) + i*sin(2*Pi*k/n) | k=0,...,n-1} under complex addition and multiplication is

1. a ring  
2. not a ring  

Question 2  (2 points)

{1,9,16,22,29,53,74,79,81} under + mod 91 and * mod 91 is

1. a ring  
2. not a ring  

Question 3  (5 points)

{0,2,4,6,8} under + mod 10 and mult mod 10 is

1. not a ring, not a field, not an integral domain  
2. a ring, not a field, not an integral domain  
3. a ring, a field, not an integral domain  
4. a ring, not a field, an integral domain  
5. a ring, a field, an integral domain  

Question 4  (5 points)

A 4Pi/3 rotation about the line connecting the vertex C to the center of the face ABD of the tetrahedron
             A
           D  B
             C
sends the vertices to (see one of the model tetrahedrons):

1.
  A
D  B
  C
 
2.
  A
C  D
  B
 
3.
  A
B  C
  D
 
4.
  C
A  B
  D
 
5.
  D
C  B
  A
 
6.
  D
B  A
  C
 
7.
  B
A  D
  C
 
8.
  B
D  C
  A
 
9.
  C
D  A
  B
 
10.
  D
A  C
  B
 
11.
  C
B  D
  A
 
12.
  B
C  A
  D
 

Question 5  (5 points)

A Pi rotation about the line connecting the midpoints of AD and BC of the tetrahedron
             A
           D  B
             C
sends the vertices to (see one of the model tetrahedrons):

1.
  A
D  B
  C
 
2.
  A
C  D
  B
 
3.
  A
B  C
  D
 
4.
  C
A  B
  D
 
5.
  D
C  B
  A
 
6.
  D
B  A
  C
 
7.
  B
A  D
  C
 
8.
  B
D  C
  A
 
9.
  C
D  A
  B
 
10.
  D
A  C
  B
 
11.
  C
B  D
  A
 
12.
  B
C  A
  D
 

Question 6  (5 points)

Match the mathematician with the example that illustrates their math.

1. Leopold Kronecker (Fundamental Theorem of Finite Abelian Groups)
  a. The quaternions are a ring that are not abelian under multiplication since ij=k, but ji=-k
2. Marjorie Lee Browne
  b. When we adjoin any root r of f(x)=x^6+6x^5+17x^4+32x^3+37x^2+26x+6 to the complex numbers C, we still get C. I.e. C(r)=C for all roots r.
3. Johann Carl Friedrich Gauss (Fundamental Theorem of Algebra)
  c. Z_6={0,1,2,...,5} under +mod6 is the direct product of
Z_2={0,1} under +mod2 and Z_3={0,1,2} under +mod3.
4. Peter Ludwig Mejdell Sylow (Sylow's First Theorem)
  d. Z_6={0,1,2,...,5} under +mod6 has
a subgroup of order 2 Z_2={0,3} under +mod6
and a subgroup of order 3 Z_3={0,2,4} under +mod6.
5. Sir William Rowan Hamilton
  e. Z_6 is a ring but not a field
6. Evariste Galois
   f. Z satisfies the ACC condition on ideals since every ascending chain of ideals terminates at some point.
7. Emmy Amalie Noether
   g. Z_3 is a finite field
8. Julius Wihelm Richard Dedekind
   h. 2x2 matrices with determinant 1 satisfying A times A transpose equals the identity form a group under matrix multiplication.
1 -->
2 -->
3 -->
4 -->
5 -->
6 -->
7 -->
8 -->