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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 . |
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.)
(x-2)-adic Expansion of a Polynomial
Expand the polynomial x^{5} - x + 1 relative to the base x - 2 in the ring of polynomials with rational coefficients.
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) EQUIV x^{2} + 2 x - 3 mod x^{2} - 3 x + 2 |
is satisfied.
(x-1)-adic Expansion of a Polynomial Fraction
Recall that rational numbers, i.e., integer ratios, have decimal expansions relative to a given base. Find the analogous expansion for the ratio of polynomials (with rational coefficients)
| {x}/{x^{2} - 3 x + 3} |
relative to (x - 1) as the polynomial base.
Do the ``digits'' repeat for this example?