All numbered problems are from the book by Dummit and Foote

HW 1: Page 4, Problems 1,2,3,4,5

HW 2: Page 11, 4, 7; Page 21, 1,2 (a)-(e), 5, 6 (d), 10.

HW 3: Page 22: 20, 21, 22, page 27, 1, 8, 9, page 32-33, 1, 3, 4, 15

HW 4: Page 34: 15, 16, 18, 19, page 40, 4, 5, 6, 8, 9.

HW 5 (Due 9/13): Page 35: 8, 11 (a,b), Page 39,40: 1 (a,b), 3, 11, 14.

HW 6 (Due 9/15): Page 35, 6, 7, Page 40: 12, 17,18

HW7 (Due 9/20): Page 52, 2,3, 5 (a,b,c), 10, 11, 14

HW8 (Due 9/22): Page 60, 12, Page 65: 1, 2, 3, 5, 6, 7.

HW9 (Due 9/27): Page 44, 3, 6, 8 (a,b), 10 (a,b)

HW10 (Due 10/6): Page 85, 1, 3, 4, 5, 7. Extra problem: What are all elements in the group of rotations O3 which have order 2016?

HW11 (Due 10/11): Page 85, 8, 9, 11, 22 (a), (b), 29.

HW 12 (Due 10/25)

Extra problems: 1. Show that if B is a basis of a vector space V, and c is any non-zero element of V, then we can replace one of the elements from B by c and still have a basis of V.

2. Show (using 1) that every two bases of the same vector space have the same number of elements.

3. Consider a vector space V of dimension n over the field Z/pZ where p is a prime. How many elements does V have?

4. Consider the vector space V of all m by n matrices over a field F. What is the dimension of V.

5. Let V be a subspace of the space of real numbers over the field of rational numbers spanned by 1, a, a^2 where a is the cube root of 2. What is the dimension of V? Consider the number b=1+a and the linear transformation of V: x maps to bx. Find the matrix of that linear transformation in the basis {1, a, a^2}.

Take home quiz, due Wednesday, 10/26.

In the following problems use a compass and a ruler (to practice, use the Web site euclidthegame.com that we used in class).

1. Given intervals of lengths a and b, construct an interval of length a+b.

2. Given intervals of lengths a and b, a > b, construct an interval of length a-b.

3. Given intervals of lengths 1, a and b construct an interval of length ab.

4. Given intervals of lengths 1, a. Construct an interval of length a1/2.

5. Given intervals of lengths 1 and a construct an interval of length 1/a.

HW 14 (Due Nov. 1) Page 311, 1 (a,b,c,d), 2 (a,b,c), 4.

HW 15 (Due Nov 3) Page 311, 5

Let a be a root of the polynomial x3 + 5x - 5. Prove that this polynomial is irreducible. Represent (1+a+a2)/(1+a 2) as a linear combination of 1, a, a2 with rational coefficients.

HW 16 (Due Nov. 8). Page 529, 530, 4, 7, 10

Show that x4+2x3+6 is irreducible. Let a be a root of that polynomial. Represent (1+a)/(1-a+a 2) as a linear combination of 1, a, a2, a3 with rational coefficients.

Let a be a root of the polynomial x2-3. Show that the polynomial x2 -a is irreducible over the field Q[a].

Let a be a root of the polynomial x3+2x+6. Find the minimal polynomial for 1+a2

HW 16 (Due Nov. 10) Page 529, 3, 14.

Let a be the square root of 2, b be the cube root of 7. Find a basis of the field Q[a,b]. What is the degree [Q[a,b]: Q[a]] ?

Let a be a root of the polynomial x3+3x+3, b be a root of the polynomial x 2 -a. What is the degree [Q[b]:Q]?

HW 17 (Due Nov. 15)

Show that using only a compass and a ruler (and a unit interval) one cannot construct an interval of length a where a is a root of the polynomial

(a) x5 - 6 x4+12

(b) x3+x2-5