Ideals generated by Polynomials and coprime polynomials (with more progress)

Solution 1:

I have a final version that is accepted by my professor:

We use double inclusion to prove $<f,g>=gcd(f,g)$

Since $K[X]$ is on Euclidean Domain, $f(x),g(x) \in K[x]$ and let $d(x)=gcd(f(x),g(x)$

Then, there exsist $h_1(x),h_2(x)$ \in K[x] such that

$$f(x)h_1(x)+g(x)h_2(x)=d(x)$$ Therefore

$$d(x)\in <f,g>$$

Then let $q(x),p(x) \in K[x]$ such that $$p(x)f(x)+q(x)g(x)=H(x)$$

Then $$f(x)=N_1(x)d(x), g(x)=N_2(x)d(x)$$

for some $N_1(x),N_2(x) \in K[X]$. Hence

$$p(x)N_1(x)d(x)+q(x)N_2(x)d(x)=H(x) \Rightarrow d(x)[p(x)N_1(x)+q(x)N_2(x)]=H(x)$$

Thus, now we have proved that $d(x)\in <f,g>$ and $d(x)$ can be the basis of any element in the ideal. Hence I is generated by $gcd(f,g)$.