PROBLEM
Let D be a principal ideal domain and $h=\gcd(f,g)$. Prove there exists $p$ and $q$ in $D$ such that $pf+gq=h$.
PROOF
We denote the ideal generated by $x$ as $(x)$. Since $D$ is a PID, we have that $(h)=(gcd(f,g))=(f)+(g)$ where $(f), (g)$ were all generated by one element. So by definition of principle idea, there exists $p,q\in D$ such that $pf=(f)$ and $qg=(g)$ so $h=pf+qg$ (up to units of $h$).
Let D be a principal ideal domain and $h=\gcd(f,g)$. Prove there exists $p$ and $q$ in $D$ such that $pf+gq=h$.
PROOF
We denote the ideal generated by $x$ as $(x)$. Since $D$ is a PID, we have that $(h)=(gcd(f,g))=(f)+(g)$ where $(f), (g)$ were all generated by one element. So by definition of principle idea, there exists $p,q\in D$ such that $pf=(f)$ and $qg=(g)$ so $h=pf+qg$ (up to units of $h$).
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