Classical Algebra (Math 326) Written Assignment

due April 14, 2000

Directions

There will be a premium placed on accuracy in the grading of this assignment. Please submit your assignment typed. If there is more than one page, please staple.
Explain your solutions.

The Analogy Between Z and F[x] when F is a field

  1. Greatest Common Divisor of Polynomials over a Finite Field

    Find the monic greatest common divisor over the finite field F_{5} of the two polynomials

    x^{4} - 1 and x^{4} - 3 x^{2} + 1 .

  2. Bezout's Identity for Polynomials

    The monic greatest common divisor of the polynomials

    f(x) = x^{5} - x + 1 and g(x) = x^{3} - x + 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. (x-2)-adic Expansion of a Polynomial

    Expand the polynomial x^{5} - x + 1 in base x - 2.

  4. Solving a Polynomial Congruence

    Determine all polynomials f(x) with rational coefficients for which the polynomial congruence

    (x^{3} + 2 x^{2} - x - 2) . f(x) = x^{2} + 2 x - 3 mod x^{2} - 3 x + 2

    is satisfied.

  5. (x-1)-adic Expansion of a Polynomial Fraction

    Recall that rational numbers, i.e., integer ratios, have decimal expansions relative to a given base. For the polynomial ratio

    {x}/{x^{2} - 3 x + 3}

    find the analogue of the decimal expansion relative to (x - 1) as a polynomial base.

    Do the coefficients repeat?

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