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].
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
Last non-zero remainder is -3
Therefore, gcd is -3
This does not seem to be correct because -3 does not divide either a(x) and b(x)
Can someone please help?
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.