quiz 4 algebraic structures
Name: DrSarah Greenwald
Start Time: Aug 13, 2000 19:11 Time Allowed: 15 min
Number of Questions: 3

Question 1  (10 points)

Match the mathematician with the example that illustrates their math.

1. Leopold Kronecker (Fundamental Theorem of Finite Abelian Groups)
  a. 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.
2. Marjorie Lee Browne
  b. Z_3 is a finite field
3. Johann Carl Friedrich Gauss (Fundamental Theorem of Algebra)

  c. 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.
4. Peter Ludwig Mejdell Sylow (Sylow's First Theorem)

  d. 2x2 matrices with determinant 1 satisfying A times A transpose equals the identity form a group under matrix multiplication.
5. Sir William Rowan Hamilton

  e. The quaternions are a ring that are not abelian under multiplication since ij=k, but ji=-k
6. Evariste Galois

  f. 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.
7. Emmy Amalie Noether

  g. Z_6 is a ring but not a field
8. Julius Wihelm Richard Dedekind

  h. Z satisfies the ACC condition on ideals since every ascending chain of ideals terminates at some point.
1 -->
2 -->
3 -->
4 -->
5 -->
6 -->
7 -->
8 -->

Question 2  (5 points)

Why isn't Z_6 a finite field?

1. Z_6 is not finite  
2. Z_6 is not a ring  
3. Z_6 is not abelian under multiplication  
4. Z_6 has no identity for multiplication  
5. Z_6 violates that all non-zero elements have multiplicative inverses in Z_6  

Question 3  (5 points)

Explain why 2 by 2 diagonal real matrices
(ie matrices of the form row1=[a,0] and row2=[0,b]
where a,b are real)
are not an integral domain, where the operations are matrix addition and matrix multiplication

1. They are not a ring  
2. They are not abelian under multiplication  
3. They do not have an identity for mult  
4. They have zero divisors