Greatest common divisor of two polynomials

I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8

================================================== ==================================

Determine the greatest common divisor of and in

and write it as a linear combination (in ) of a(x) and b(x).

================================================== ===================================

In working on this I applied the Division Algorithm to a(x) and b(x) resulting in

then

Last non-zero remainder is -3

Therefore, gcd is -3

BUT!

This does not seem to be correct because -3 does not divide either a(x) and b(x)

Can someone please help?

Peter

Re: Greatest common divisor of two polynomials

Hi,

I think your plan of attack is exactly correct. Remember you are working in Q[x], not Z[x]. So in Q[x], -3 __does__ divide any polynomial; e.g. . That is -3 is a unit in Q[x].

Re: Greatest common divisor of two polynomials

Using the rational roots theorem, you can check that has no rational roots. I pretty sure this means it has no (non-unit) factors in . This would make irreducible. Therefor, I believe , any unit in , i.e. any (non-zero) constant polynomial. Since there ordering on is only concerned with the polynomials degree, I don't think any constant polynomial would be called "greater" then any other.