I'm meant to find the greatest common divisor of polynomials $x^3+2x+1$ and $2x^2+3$ and their respective Bezout coefficients in fields $Q[x]$ and $Z5[x]$.
Quite lost I start the algorithm for $Q[x]$ getting from $x^3+2x+1=(2x^2+3)(1/2x) + (1/2x+1) \rightarrow (2x^2+3)=(1/2x+1)(4x-8) + (11)$ which is where I stopped.
I'm sure this is rather basic but how do I deal with the situation now as I have only been given numerical exercises.
As for the Z5 I'm even more confused since I expected an easier solution but have gotten $x^3+2x+1=(2x^2+3)(3x)+(3x+1) \rightarrow (2x^2+3)=(3x+1)(4x+2) + 1.$
Which would lead me to conclusion that GCD of both is 1 but am unable to confirm/prove it.
Thanks for your time