1. ## unit

Suppose $\displaystyle f(x)$ is relatively prime to $\displaystyle 0_F$. Then $\displaystyle \gcd(f(x), 0_F) = 1_F$. So $\displaystyle 1_F = f(x)u(x)+0_{F}v(x)$ for some polynomials $\displaystyle u(x)$ and $\displaystyle v(x)$. Thus $\displaystyle 1_F = f(x)u(x)$.

So this means that $\displaystyle f(x)$ is a unit?

2. Originally Posted by Sampras
Suppose $\displaystyle f(x)$ is relatively prime to $\displaystyle 0_F$.
this never happens! every polynomial obviously divides the zero polynomial. \

EDIT: see my next post!

3. Originally Posted by NonCommAlg
this never happens! every polynomial obviously divides the zero polynomial.
So since the hypothesis is false, what can be said about $\displaystyle f(x)$? It is not a polynomial?

4. Originally Posted by Sampras
So since the hypothesis is false, what can be said about $\displaystyle f(x)$? It is not a polynomial?
ok, i was too quick in responding i guess ... sorry about that! as i said, every polynomial divides the zero polynomial and thus $\displaystyle \gcd(0,f(x))=f(x),$ for any polynomial. so $\displaystyle \gcd(0,f(x))=1$ if and

only if $\displaystyle f(x)$ and $\displaystyle g(x)=1$ are associates, i.e. if and only if $\displaystyle f(x)$ is a unit. so your conclusion is right.