Relation between generator polynomial and codeword in a cyclic code

Question

I'm trying to solve the following exercise, but I can't use an hypothesis.

Let $F$ be a finite field and $a(x)$ a poly of degree $n$ over $F[x]$. Let $C$ the smallest cyclic code of length $n$ over $F$ with $a(x)$ as codeword, and let $g(x)$ the generator poly. Show that $g(x)=\text{G.C.D} (a(x),x^n-1)$

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By definition, I know that $g|x^n - 1$ and $g$ has degree $n-k$. Also, since $a(x)$ is codeword, then $a(x)=f(x) g(x)$, for some $f(x)$ of degree $n-k$, and hence $g|a(x)$.

Now I'm supposed to use the fact that $C$ is the smallest cyclic code that contains $a$ and that $a$ has degree $n$, but I really can't figure it out how two use it properly.

Answer 1

(By the way, I think the original problem is misstated. I think they mean "the degree of $a(x)$ is less than $n$.)

This is all just the picture that cyclic codes of length $n$ correspond to ideals in $F[x]/(x^n-1)$, and the fact that $F[x]$ is a principal ideal domain.

The interpretation of the GCD in a PID is exactly: $g=\gcd(a,b)$ if and only if $(g)$ is the smallest ideal containing both $(a)$ and $(b)$.

And so you have it:

$(g(x))$ is the smallest ideal containing $(a(x))$ and $(x^n-1)$,

but this is also just saying

$(g(x))/(x^n-1)=(a(x), x^n-1)/(x^n-1)$.

So $(g(x))/(x^n-1)$ is the smallest cyclic code of length $n$ containing the word corresponding to $a(x)$.