Showing that there exists a non constant Polynomial over a finite field without roots.

Question

I am trying to show the following statement:

Let F be a finite field. Show that there exists a non-constant polynomial $f \in F[x]$ that has no roots in F.

My idea was to simply construct such a polynomial since this is only an existence proof. But my solution seems to be too banal. Below is what I came up with:

Let $g$ be a polynomial in F with $g(x) = 0 \forall x\in F$. We can now simply add a constant factor from F to g s.t

$g_{1}(x) = g(x) + a$ where $a$ is constant $a\in F$. It is now simple to see that $g_{1}(x) \neq 0 \forall x\in F$.

Is this right? It seems to me that $g_{1}(x)$ is just a constant in this case and not a polynomial, hence the proof is wrong... some help and tipps will be much appreciated.

thank you

Answer 1

You can proceed like this:

Let $p(x)=\prod_{a\in F}(x-a)$. As $F$ is a finite field, it contains $F_p$ for some $p\in \mathbb P$. Pick $\overline 1\in F_p\subset F$. The polynomial $$q(x)=p(x)+\overline 1$$ can't have any roots in $F$.