Let K be a field, $a,b \in K[X]$ and $i,j \in K$ with $i \not= 0$ and $j \not= 0$.
Prove that $gcd(a,b) = gcd(a*i,b*j)$
Could you please help me proving this. I have the following idea:
For any common divisor $d$ of $a$ and $b$ is: $d | a$ and $d | b$ and $d | a*i$ and $d | b*j$
For any common divisor $d$ of $a*i$ and $b*i$ is: $d | a$ and $d | b$ and $d | a*i$ and $d | b*j$
Thus every common divisor of a and b is an common divisor of $a*i$ and $b*j$. But how to prove it is the biggest divisor? Thanks!
Generally, in any gcd domain, if $\,i,j\,$ are units (invertibles) then
$$ d\mid a,b \iff d\mid ai,bj$$
Thus $\,a,b\,$ and $\,ai,bi\,$ have the same set $\,S\,$ of common divisors $\,d,\,$ therefore they also have the same greatest common divisor - any greatest element of $S\,$ (= greatest degree polynomial in your case, usually normalized to be monic, i.e. have leading coefficient $=1)$,