Math 361, Spring 2022, Assignment 10
From cartan.math.umb.edu
Carefully define the following terms, and give one example and one non-example of each:[edit]
- Standard representative (of an element of F[x]/⟨m⟩; i.e. the representative whose uniqueness is guaranteed by the theorem concerning unique representation below).
- Standard generator (of F[x]/⟨m⟩; usually this is denoted by α).
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem concerning unique representation of elements of F[x]/⟨m⟩.
- Theorem concerning m(α) (where α is the standard generator of F[x]/⟨m⟩).
Carefully describe the following procedures:[edit]
- Procedure to calculate the standard representation of the product (f+⟨m⟩)(g+⟨m⟩) (i.e. the "machine implementation" of multiplication in F[x]/⟨m⟩).
- Procedure to rewrite "high" powers of the standard generator α in terms of lower powers, using the theorem concerning m(α) (i.e. the "human implementation" of multiplication in F[x]/⟨m⟩)).
Solve the following problems:[edit]
- Let R denote the quotient ring Z2[x]/⟨x2+1⟩. List the elements of R, then make a multiplication table. Is R a field?
- Let GF(8) denote the quotient ring Z2[x]/⟨x3+x+1⟩. List the elements of GF(8). Be sure to list each element only once. (You will probably find it more pleasant to write them in terms of the standard generator α rather than using coset notation.)
- Working in GF(8), compute the sum (1+α2)+(1+α).
- Using the "machine implementation" of multiplication in GF(8), compute the product (1+α2)(1+α). Be sure to write your answer in its standard representation.
- Working in GF(8), find a formula for α3 in terms of lower powers of α. (Hint: use the theorem regarding m(α).)
- Use the formula you found above to compute the standard representations of α4,α5, α6, and α7.
- Redo your calculation of (1+α2)(1+α), this time avoiding the "machine implementation" in favor of the formula you found above for α3. Verify that you obtain the same answer.
- Suppose m∈Zp[x] is a polynomial of degree d. Compute the cardinality of the ring Zp[x]/⟨m⟩. (Hint: use the theorem on unique representation of elements. How many choices are there for each coefficient, and how many coefficients are there?)
- Verify that the formula you found above correctly predicts the number of elements of GF(8).
Questions:[edit]
Solutions:[edit]
Definitions and Theorems:[edit]
https://drive.google.com/file/d/12qg0ua83mJJKQduzpgLjo7SsZ2KGwvEl/view?usp=sharing
Problems:[edit]
https://drive.google.com/file/d/1sWc5CPO9jzdwHJEcU71iZCacFi1Sm7jf/view?usp=sharing