Abstract Algebra: Theory and Applications · Problem 17.8.4

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · polynomial-gcd

Problem

Define the polynomials

p = x^3 + 2x^2 + 2x + 4

and

q = x^4 + 2x^2

as polynomials with coefficients from the integers. Compute

\gcd(p, q)

and verify that the result divides both

p

and

q

(just form a fraction in Sage and see that it simplifies cleanly, or use the .quo\_rem() method). Proposition 17.2.5 says there are polynomials

r(x)

and

s(x)

such that the greatest common divisor equals

r(x)p(x) + s(x)q(x)

, if the coefficients come from a field. Since here we have two polynomials over the integers, investigate the results returned by Sage for the extended gcd,

\texttt{xgcd(p, q)}

. In particular, show that the first result of the returned triple is a multiple of the gcd. Then verify the ``linear combination'' property of the result.

Browse the problem library