Let $g(x)$ be the generator polynomial of a binary cyclic code $C$ and let $h(x)=\left(x^{n}-1\right)/g(x)$ be the parity check polynomial. Show that if $a(x)$ is a polynomial satisfying $\gcd\left(a(x), h(x)\right)=1$, then $a(x)g(x)$ is a generator of $C$.
Now what I am not understanding is why I must show that $a\left(x\right)$ will have to satisfy $\gcd\left(a(x), h(x)\right)=1$. As long as $a(x)$ is of least degree and a divisor of $x^n -1$, the polynomial $a(x)g(x)$ should be a generator of $C$, correct?
As always, thanks for the help.
If the gcd condition does not hold you won’t obtain the same ideal as $C=<g(x)>.$