3110 Final Exam Login ID: drsarah Attempt: 1 / 1 Max. Score: 60 Quiz Started: May 08, 2000 8:46 Quiz Finished: May 08, 2000 9:00 Time Spent: 14 min., 35 sec. Student finished 1 hr., 15 min., 25 sec. ahead of the 90 min. time limit.

### Question 1  (9 points)

Match the equation to the corresponding functional qualities.

 y=cos(x) y is 1-1 and onto y=e^x y is not a function y=x^3-x y is onto but not 1-1 y=x y is 1-1 but not onto x=y^2 y is neither 1-1 nor onto

Student Response:

 y=cos(x) --> y is neither 1-1 nor onto Correct y=e^x --> y is 1-1 but not onto Correct y=x^3-x --> y is onto but not 1-1 Correct y=x --> y is 1-1 and onto Correct x=y^2 --> y is not a function Correct

Score: 9.0 / 9.0

### Question 2  (9 points)

Match the polynomials to the corresponding statements.

 x^5-2x^3-8x-2 If at all reducible, then solvable by radicals x^5 always solvable by radicals x^5-1 partially reducible, solvable by radicals 5th degree polynomials completely reducible, solvable by radicals 4th or lower degree polynomials irreducible, not solvable by radicals

Student Response:

 x^5-2x^3-8x-2 --> irreducible, not solvable by radicals Correct x^5 --> completely reducible, solvable by radicals Correct x^5-1 --> partially reducible, solvable by radicals Correct 5th degree polynomials --> If at all reducible, then solvable by radicals Correct 4th or lower degree polynomials --> always solvable by radicals Correct

Score: 9.0 / 9.0

### Question 3  (9 points)

Match the examples to their algebraic structures.

 2Z integral domain, not a field quaternions integral domain and field unit quaternions ring which is abelian under *, has a mult id, but is not an integral domain symetries of tetrahedron ring which is abelian under *, has no zero divisors, but is not an integral domain Z_6 non-abelian group, not a ring Z ring which has no zero divisors, has a mult id, but is not an integral domain Z_p, where p is prime abelian group, not a ring

Student Response:

 2Z --> ring which is abelian under *, has no zero divisors, but is not an integral domain Correct quaternions --> ring which has no zero divisors, has a mult id, but is not an integral domain Correct unit quaternions --> non-abelian group, not a ring Correct symetries of tetrahedron --> abelian group, not a ring Correct Z_6 --> ring which is abelian under *, has a mult id, but is not an integral domain Correct Z --> integral domain, not a field Correct Z_p, where p is prime --> integral domain and field Correct

Score: 9.0 / 9.0

### Question 4  (14 points)

I've been very sloppy in my proofwriting and left out the first sentance which will tell you what I'm trying to prove, and the last sentance which tells you what I've proven, but at least I've included a list of statments with my proofs.

Match the statements on the left with the incomplete proof on the right that (when completed) proves or disproves the statement.

 a,b,c in R, a not 0, ab=ac implies b=c Let R=Z_6={0,1,2,3,4,5}. Take a=0, b=1. Notice that a,b in R, ab=0*1=0 and a=0, as desired. a,b, in R, ab=0 implies a=0 or b=0 Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=1, c=4. Notice that a,b,c in R and a not 0. Also, ab=2*1=2=2mod6=8mod6=2*4=ac, but 1 is not equal to 4, as desired. a in R implies, a not 0, a^2 = a implies a=0 or a=1. Let R=Z_6={0,1,2,3,4,5}. Take a=4. Notice a in R, a not 0, a^2=4^2=16=4mod6 and a not 0 and a not 1, as desired. There exists R, there exists a,b, in R s.t. ab=0 and (a=0 or b=0) Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=1, c=1. Notice a,b,c in R, a not 0, ab=2*1=2=2*1=ac and b=1=c, as desired. There exists R, there exists a in R s.t. a not 0, a^2=a and (a=0 or a=1) Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=3. Notice that a,b in R and ab=2*3=6=0mod6 and 2 is not 0 and 3 is not 0, as desired. There exists R, there exists a,b,c in R s.t. a not 0, ab=ac and b=c. Let R=Z_6={0,1,2,3,4,5}. Take a=1. Notice a in R, a not 0, a^2=1^2=1, and a=1, as desired.

Student Response:

 a,b,c in R, a not 0, ab=ac implies b=c --> Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=1, c=4. Notice that a,b,c in R and a not 0. Also, ab=2*1=2=2mod6=8mod6=2*4=ac, but 1 is not equal to 4, as desired. Correct a,b, in R, ab=0 implies a=0 or b=0 --> Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=3. Notice that a,b in R and ab=2*3=6=0mod6 and 2 is not 0 and 3 is not 0, as desired. Correct a in R implies, a not 0, a^2 = a implies a=0 or a=1. --> Let R=Z_6={0,1,2,3,4,5}. Take a=4. Notice a in R, a not 0, a^2=4^2=16=4mod6 and a not 0 and a not 1, as desired. Correct There exists R, there exists a,b, in R s.t. ab=0 and (a=0 or b=0) --> Let R=Z_6={0,1,2,3,4,5}. Take a=0, b=1. Notice that a,b in R, ab=0*1=0 and a=0, as desired. Correct There exists R, there exists a in R s.t. a not 0, a^2=a and (a=0 or a=1) --> Let R=Z_6={0,1,2,3,4,5}. Take a=1. Notice a in R, a not 0, a^2=1^2=1, and a=1, as desired. Correct There exists R, there exists a,b,c in R s.t. a not 0, ab=ac and b=c. --> Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=1, c=1. Notice a,b,c in R, a not 0, ab=2*1=2=2*1=ac and b=1=c, as desired. Correct

Score: 14.0 / 14.0

### Question 5  (5 points)

A rotation of 2Pi/3 about the line through A and the center of the face DCB
FOLLOWED BY a Pi rotation about the line connecting the midpoints of AB and CD of the tetrahedron
A
D  B
C
sends the vertices to (see one of the model tetrahedrons):

 0.0% 1 A D  B   C 0.0% 2 A C  D   B 0.0% 3 A B  C   D 0.0% 4 C A  B   D 0.0% 5 D C  B   A 100.0% 6 D B  A   C 0.0% 7 B A  D   C 0.0% 8 B D  C   A 0.0% 9 C D  A   B 0.0% 10 D A  C   B 0.0% 11 C B  D   A 0.0% 12 B C  A   D

Score: 5.0 / 5.0

### Question 6  ( points)

Match the mathematician with the example that illustrates their math.

 1. Niels Abel a. proved that x^5+y^5=z^5 has no non-zero whole number solutions. 2. Marjorie Lee Browne b. The quaternions are a ring that are not abelian under multiplication since ij=k, but ji=-k 3. Julius Wihelm Richard Dedekind c. 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. 4. Leonhard Euler d. Proved the Epsilon Conjecture which says that Taniyama-Shimura implies Fermat's Last Theorem 5. Pierre de Fermat e. mentions the equation 8x8x8x8=4 in the cut "As" from Songs in the Key of Life. 6. Lodovico Ferrari f. Z_3 is a finite field 7. Evariste Galois g. proved that x^3+y^3=z^3 has no non-zero whole number solutions. 8. Johann Carl Friedrich Gauss (Fundamental Theorem of Algebra) h. Z_6 is a ring but not a field 9. Sophie Germain i. 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. 10. Sir William Rowan Hamilton j. Proved Fermat's Last Theorem by proving Taniyama-Shimura 11. Leopold Kronecker (Fundamental Theorem of Finite Abelian Groups) k. proved that x^4+y^4=z^4 has no non-zero whole number solutions. 12. Adrien-Marie Legendre l. the circle is a manifold and a group under complex multiplication of norm 1 13. Lie m. Worked on the solution of a quartic by radicals 14. Francisco Maurolycus n. Worked on the solution of a quintic by radicals 15. Emmy Amalie Noether o. Z satisfies the ACC condition on ideals since every ascending chain of ideals terminates at some point. 16. Ken Ribet p. First mathematician to examine a general approach (for all powers) for Fermat's Last Theorem 17. Peter Ludwig Mejdell Sylow (Sylow's First Theorem) q. the first mathematician known to use induction (in the proof that for all n in N, 1+3+5+...+(2n-1)=n^2. 18. Andrew Wiles r. 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. 19. Stevie Wonder s. 2x2 matrices with determinant 1 satisfying A times A transpose equals the identity form a group under matrix multiplication.

Student Response:

 1 --> n Correct 2 --> s Correct 3 --> h Correct 4 --> g Correct 5 --> k Correct 6 --> m Correct 7 --> f Correct 8 --> c Correct 9 --> p Correct 10 --> b Correct 11 --> i Correct 12 --> a Correct 13 --> l Correct 14 --> q Correct 15 --> o Correct 16 --> d Correct 17 --> r Correct 18 --> j Correct 19 --> e Correct

Score: 100%