Classical Algebra

Written Assignment No. 5

due Friday, December 7, 2007

Directions

Written assignments must be typeset.

While it is neither necessary nor desirable to show small details of computation, you must indicate what you are doing, give major steps in computation, and explain any reasoning used.

Accuracy is important. With only 10 points for this assignment, there is limited room for partial credit on a problem.

Please remember that no collaboration is permitted on this assignment.

No late submission of this assignment will be allowed.

Problems

  1. (1 point) Find the monic greatest common divisor over the finite field F5 of the two polynomials x41andx43x2+1.

  2. (1 point) The monic greatest common divisor of the polynomials fx=x5x+1andgx=x3x+1, regarded as polynomials with rational coefficients, is the constant polynomial 1. Express 1 as a polynomial linear combination of f and g. (Be sure to verify the correctness of your answer by expanding the linear combination.)

  3. Find the order of the congruence class of the polynomial fx modulo the polynomial mx when the field of coefficents is Fp in the following cases:

    1. (1 point) fx=x, mx=x2+1, and p=5.

    2. (1 point) fx=x, mx=x2x+1, and p=5.

    3. (1 point) fx=x2, mx=x2+5x+1, and p=7.

  4. (1 point) Find a polynomial ft in F5[t] whose congruence class modulo mt is a primitive element for the field F5[t]mtF5[t] when mt=t2t+1.

  5. (1 point) Write a proof of the following proposition: If F is a field and ft is in the ring F[t] of polynomials with coefficients in F, then the polynomial t and the polynomial ft have no (non-constant) common factor if and only if f00.

  6. F4 is defined to be the field F2[t]t2+t+1F2[t].

    1. (1 point) How many congruence classes are there of polynomials in F4[x] modulo the polynomial x3+x+1 ?

    2. (1 point) Explain why the polynomial x3+x+1 is irreducible over F4.

    3. (1 point) Find a primitive element for the ring F4[x]x3+x+1F4[x] of congruence classes.