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.
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### 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