# unit

• Sep 28th 2009, 03:41 AM
Sampras
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?
• Sep 28th 2009, 03:50 AM
NonCommAlg
Quote:

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!
• Sep 28th 2009, 03:52 AM
Sampras
Quote:

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?
• Sep 28th 2009, 04:00 AM
NonCommAlg
Quote:

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.