polynomial units in field 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.
 A: It is not true that if $p$ is reducible then it is a unit. Recall that an element $p\in F[X]$ is a unit if there exists $q\in F[X]$ such that $pq=1$. If $F$ is a field, then comparing degrees shows that all units are nonzero constants. Conversely all nonzero constants are clearly units.
Now consider for example the polynomial $p=X^2\in F[X]$, which is not a unit. Then for the polynomials $f=g=X\in F[X]$ we have $p=fg$, and neither $f$ nor $g$ is a unit. This means $p$ is reducible, but not a unit.
The same reasoning shows that more generally, a product of two nonconstant polynomials in $F[X]$ is reducible but not a unit.
A: Here $F$ is assumed to be a field for the purpose of your Theorem.  In that case the units of $F[X]$ are the nonzero constants (i.e. units of $F$), where $X$ is assumed to be an indeterminate.
More generally if $F$ is an integral domain (no zero divisors), then the units of $F[X]$ are again the units of $F$ (considered as degree zero polynomials).
The point of defining "irreducibles" to exclude units is to give a definition that is widely applicable.  We could have said, apropos of a polynomial ring, that the degree should be nonzero, but the way stated applies to "irreducibles" in any integral domain.
For univariate polynomials with coefficients over a field, an irreducible polynomial is a nonzero polynomial that cannot be factored except as a unit times another polynomial.  This also generalizes to polynomials in more than one variable.
