There are a couple questions of the form: let $I$ be an ideal in the given Euclidean Domain, $D$. Find a generator for $I$ given that it is generated by $a,b\in D$. More specifically, we might have:
Let $I$ be the ideal generated by $x^3+x^2-2x-2$ and $x^3-x^2-2x+1$ in $\mathbb{Q}[x]$. My idea is to find the $\gcd$ of the two polynomials and the ideal should be principal, generated by that polynomial. Since $x^3+x^2-2x-2=(x^2-2)(x+1)$ and neither divide $x^3-x^2-2x+1$, we have that $\gcd=1\implies$ the ideal is $\mathbb{Q}[x]$, itself. First, is this correct?
Second, how do we extrapolate division with remainder to other Euclidean Domains, say $\mathbb{Z}[i]$, the Gaussian integers?
More specifically, how might I find the $\gcd$ of $5+5i$ and $3-i$?
actually $$ (1+2i)(3-i) = 5 + 5i $$
Simple continued fraction tableaux for ring Q[x]: $$ \begin{array}{cccccccccc} & & 1 & & \frac{x-1}{2} & & -4x+4 & & \frac{x+1}{2} & \\ \frac{0}{1} & \frac{1}{0} & & \frac{1}{1} & & \frac{\frac{x+1}{2}}{\frac{x-1}{2}} & & \frac{-2x^2 + 3}{-2x^2 + 4x-1} & & \frac{-x^3 - x^2 + 2x + 2}{-x^3 + x^2 +2x-1} \end{array} $$ so $$ (x^3 + x^2 -2x-2)(-2x^2 + 4x - 1) - (x^3 - x^2 - 2x + 1)(-2x^2 + 3)= -1 $$