Let $F$ be any field and $a,b\in F,\,\,a\neq b$. Find the greatest common divisor of $f(x) = x + a$ and $g(x) = x + b$.
Since the degree of both is $1$, the gcd is $1$ or $f(x)$ or $g(x)$, since $a\neq b$. So $\gcd(f(x),g(x))=1$.
Am I right for the answer and proving?
Let $a \sim b$ denote that $a$ is associated with $b.$ The only divisors up to associates of $x+a$ are $1$ and $x+a$, and the only divisors of $x+b$ up to associates are $1$ and $x+b.$ Since $a\neq b$, $x+a$ is not associated to $x+b$ so the greatest common divisor is the only common divisor, $1.$
Hint $\rm\,\ gcd(x\!+\!a,f(x))\, =\, x\!+\!a\:$ if $\rm\:f(-a)=0,\ $ else $1,\,\:$ by $\rm\ mod\ x\!+\!a\!:\,\ x\equiv -a\:\Rightarrow\: f(x)\equiv f(-a).$
