Math 306, Spring 2012
First Midterm Exam
Name:
Student ID:
Directions: Check that your test has 9 pages, including this one and the blank one on the bottom
(which you can use as scratch paper or to continue writing out a solution if you run out room elsew
Math 306, Spring 2012
Final Exam Solutions
Name:
Student ID:
Directions: Check that your test has 12 pages, including this one and the blank one on the bottom
(which you can use as scratch paper or to continue writing out a solution if you run out room el
Math 306, Spring 2012
Homework 4, due Friday, February 24
(1) Find the minimal polynomials over the smaller eld of the following elements in the given extensions.
(a) in C : Q
3i
(b) 18 in R : Q
(c) e in C : Q(e)
(d) 5+1 in C : Q
2
(e) e2i/11 in C : Q
(2)
Math 306, Spring 2012
Homework 5, due Friday, March 2
(1) (a) (3 pts) Find a linearly dependent set of three vectors in R3 , but such that any set of two of them is linearly
independent.
(b) (5 pts) Let V be a vector space over C. Suppose that B = cfw_v1
Math 306, Spring 2012
Homework 3, due Friday, February 17
(1) Each of the following statements is false. Disprove each of them by providing a counterexample or by appealing
to the denitions.
(a) If K is a eld, every polynomial in K[t] has a root in K (rec
Math 306, Spring 2012
Final Exam
Name:
Student ID:
Directions: Check that your test has 11 pages, including this one and the blank one on the bottom
(which you can use as scratch paper or to continue writing out a solution if you run out room elsewhere).
Math 306, Spring 2012
Homework 1, due Friday, February 3
(1) Each of the following statements is false. Disprove each one by providing a counterexample or by appealing to
the denition.
(a) If R is a ring, then for every nonzero f, g R[t], we have deg(f +
Math 306, Spring 2012
Second Midterm Exam, April 12, 2012
Name:
Student ID:
Directions: Check that your test has 8 pages, including this one and the blank one on the bottom
(which you can use as scratch paper or to continue writing out a solution if you r
Math 306, Spring 2012
First Midterm Exam Solutions
Name:
Student ID:
Directions: Check that your test has 9 pages, including this one and the blank one on the bottom
(which you can use as scratch paper or to continue writing out a solution if you run out
Math 306, Spring 2012
Second Midterm Exam Solutions
Name:
Student ID:
Directions: Check that your test has 8 pages, including this one and the blank one on the bottom
(which you can use as scratch paper or to continue writing out a solution if you run out
Math 306, Spring 2012
Homework 6, due Friday, March 16
(1) (5 pts/part)
(a) Prove that cos(2/5) = 51 . (Hint: Using the equation (cos(2/5) + i sin(2/5)5 = 1, rst show that
4
= cos(2/5) is a root of 16x5 20x3 + 5x 1, which factors into a linear piece time
Math 306, Spring 2012
Homework 2, due Friday, February 10
(1) (5 pts) Suppose that R is an integral domain that is not a eld. Prove that R[x] is not a principal ideal domain.
(Hint: Let c R be nonzero and noninvertible and consider I = (c, x).)
(2) (5 pts