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 --> Choose Match abcdefgh 2 --> Choose Match abcdefgh 3 --> Choose Match abcdefgh 4 --> Choose Match abcdefgh 5 --> Choose Match abcdefgh 6 --> Choose Match abcdefgh 7 --> Choose Match abcdefgh 8 --> Choose Match abcdefgh