I am new on this topic and would like to have some clarifications.
We were given a definition and theorem as follows:
Definition : A polynomial $p$ in $F[x]$ is irreducible if $p$ is not a unit of $F[x]$, and if $p = f\cdot g$ then either $f$ or $g$ must be a unit.
By my own interpretation if it is reducible then $p$ must be a unit of $F[x]$.
Theorem: If $F$ is a field then the only units of $F[x]$, that is polynomials $p$ such that exists $q$, $p\cdot q = 1$ and in ($F[x]$), are the units of $F$. Thus it can only be constant polynomial of degree $=0$.
Then we have this fact $\dots$ $(x^2)+1$ is irreducible in $ℤ/3ℤ$ (but not in $ℤ/5ℤ$!)
Now my professor said it is irreducible because it cannot be factored in $ℤ/3ℤ$ but can in $ℤ/5ℤ$ which is $(x+3)\cdot (x+2)$. I am confused because the theorem stated that units of $F$ are only constant numbers. However we have $(x^2)+1$ which is reducible in $ℤ/5ℤ$ and thus is a unit in $ℤ/5ℤ$???
I am so confused, I don't know where or what I am doing wrong.