Question 1  (9 points)

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)

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)

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)

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)

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)

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%

Total Score: 46.0 / 46 = 100.0%