## Fall 2011

There are 3 kinds of problems assigned.

Basic Problems: Don't hand these problems in. You should do them before the Core Problems in the same section, to help your understanding.

Core Problems: Everyone must turn these problems in. Always justify your answer, even if the question does not explicitly say so.

Challenge Problems: MATH 6000 students must turn at least one of these per week. They are extra credit for MATH 4000 students. Please hand these in stapled separately from the other problems.

Assignment 1: Due Thursday 8/25/11

Basic Problems: Sec. 1.1 # 3

Core Problems: Sec. 1.1 # 4 fgh, 6 b (use a in showing b), 7, 11

Challenge Problems: Sec. 1.1 # 2 (for full credit, you must figure out the generalization), 8, 17, 19

Assignment 2: Due Thursday 9/1/11

Basic Problems: Sec. 1.2 # 1bcd, 6

Core Problems: Sec. 1.2 # 2, 7, 12, 13

Challenge Problems: Sec. 1.2 # 16, 17, 19, 20

Assignment 3: Due Thursday 9/8/11

Basic Problems: Sec. 1.3 # 5, 20bd

Core Problems: Sec. 1.3 # 8, 9, 14, 20c, 21c, 29

Challenge Problems: Sec. 1.3 # 34, 36, 37, 39

Assignment 4: Due Thursday 9/15/11

Basic Problems: Sec. 1.4 # 1 (for Z_7), 3

Core Problems: Sec. 1.4 # 6bcd, 7, 9, 12, 13

Challenge Problems: Sec. 1.4 # 16, 17, 18, 19

Assignment 5: Due Friday 9/23/11

Note: The due date is 1 day later than usual because Test 1 is Thursday 9/22/11.x

Basic Problems: Section 2.3 # 2, 8

Core Problems: Section 2.3 # 6e, 9bcg, 13, 19. Comments: On 6e, don't repeatedly use the addition formulas for sin and cos; instead think of a way to use deMoivre's formula. For 9b, it may help to look at Example 9 on p. 63 (note that this works even if the angle isn't nice). For 9c, it may be helpful to replace an angle by that angle plus a multiple of 2 pi.

Challenge Problems: Section 2.3 # 14, 21

Assignment 6: Due Thursday 9/29/11

Basic Problems: Section 2.4 # 1af

Core Problems: Section 2.4 # 2, 6b, 8, 9. Note: For 6b, use the method of proof of the cubic formula as we did in class, rather than the formula itself -- and don't just plug in the answers given!

Challenge Problem: Section 2.4 # 10

Assignment 7: Due Thursday 10/6/11

Basic Problems: Section 3.1 # 1bde, 10 abc

Core Problems: Section 3.1 # 2cd, 6, 10def, 13, 15. (Hint for #10: there is an easy way to tell if a polynomial with Z_p coefficients has a root. So this can tell you if a cubic polynomial is irreducible (why?). For a fourth degree polynomial, you need to check whether there is a root, and whether it factors as 2 quadratic polynomials.

Challenge Problems: Section 3.1 # 20, 23

Assignment 7: Due Thursday 10/13/11

Basic Problems: Section 3.2 # 1, 5c

Core Problems: Section 3.2 # 2b, 3ace, 6c, 11, 14

Challenge Problems: Section 3.2 # 16, 17, 18.

Comments: For problems involving splitting fields, you'll need to read the definition and Ex. 4 on p. 99-100. (Intuitively, the splitting field of a polynomial is the field you get by adjoining the roots of the polynomial. So the splitting field of x^2 + 1 is Q[i] since the roots are plus or minus i.) For #16, you can use the fact that pi is not the root of any polynomial with rational coefficients (we say pi is "transcendental").

Assignment 8: Due Thursday 10/20/11

Basic Problems: Sec. 3.3 #3

Core Problems: Sec. 3.3 # 2 abcdfi, 4, 5, 6, 7, 8

Challenge Problem: Sec. 3.3 # 9

Assignment 9: Due Thursday 10/27/11

Basic Problems: Sec. 5.1 #3

Core Problems: Sec. 5.1 # 5, 9, 10, 11bcdef, 13, 15, 20

Challenge Problems: Sec. 5.1 # 21, 22, 23

Note: I have deleted 11g since it relies on Ch. 4.

Assignment 10: Due Thursday 11/3/11

Basic Problems: Sec. 4.1 # 2

Core Problems: Sec. 4.1 # 3, 4bcdf, 6, 8, 15, 17

Challenge Problems: 18, 21

Assignment 10: Due Friday 11/11/11 (Remember, Test 2: Thursday 11/10/11!)

Core Problems: Section 4.1 #10, 14 ab ; Section 4.2 # 3 abcde, 11abc, 13 abc

Challenge Problems: Section 4.2 # 13de, 18, 19

Note: For Section 4.2 # 3, 11 try to use the Fundamental Homomorphism Theorem. For 11, you'll need the definition of a direct product (p. 127 -- I'll discuss on Tuesday).

Assignment 11: Due Thursday 12/1/11

Basic Problems: Section 5.2 # 2, 5

Core Problems: Section 5.2 # 1, 9, 10, 11, 13

Challenge Problem: Section 5.2 # 16

Final Assignment (not to be turned in, but to prepare for Final Exam:

Section 5.3 # 1, 2, 3a, 5. (For #5, the Frobenius automorphism is defined on p. 167 as the map sigma: F ---> F given by sigma(a) = a^p.)

Good luck!