Finding the units in the ring F[x]

[hmmmmm]

I have to find all the units in the ring $F[x]$ where $F$ is a field.

Clearly all polynomials of degree 0 are units as they are in the field F.

Now suppose $\alpha(x)\beta(x)=1$ which gives $\mbox{deg}(\alpha(x))=-\mbox{deg}(\beta(x))$ which gives $\mbox{deg}(\beta(x)=\mbox{deg}(\alpha(x)=0$ so the units are precisely the polynomials of degree.

Is this correct?

Thanks for any help

 

[Opalg]

Re: Finding the units in the ring $F[x]$

That is correct. The units are the zero-degree polynomials (the nonzero elements of the field).