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} − 3x^{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} + 2x^{2} − x − 2) · f(x) ≡ x^{2} + 2x − 3 mod x^{2} − 3x + 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

\frac{x}{x^{2} − 3x + 3}

find the analogue of the decimal expansion relative to (x − 1) as a polynomial base. Do the coefficients repeat?

Document network location:

DVI: http://math.albany.edu:8000/math/pers/hammond/course/mat326s2000/assgt/cla000407.dvi
HTML: http://math.albany.edu:8000/math/pers/hammond/course/mat326s2000/assgt/cla000407.html