How to check if a polynomial in $F[x]$ is irreducible or not, for $F$ a finite field

Question

I need to check the irreducibility of $p(x) \in F[x]$, where $F$ is a finite field. I have read and checked on several exercises on the internet. Their solutions are as follows:

For instance, let $p(x)$ an arbitrary polynomial in $\mathbb{Z}_5[x]$.

If $p(x)$ has no zeros in $\mathbb{Z}_5$, then they say that $p(x)$ is an irreducible polynomial in $\mathbb{Z}_5[x]$.

I am confused at this point: The polynomial $p(x)=(x^2+2)(x^2+3)$ has no zeros in $\mathbb{Z}_5[x]$, but it is reducible? Where is my mistake?

Answer 1

In $Z_5[x]$ the polynomial $p(x) = x^4 + 1$. This can be checked to have no zeroes in your field.

It seems the definition you are using is incorrect. Reducible means factorable into polynomials of lesser degree.

Answer 2

Let $D$ be an integral domain with unity, a polynomial $f(x) ∈ D[x]$ such that $deg(f(x)) ≥ 1$ is irreducible polynomial in $D[x]$, if whenever $f(x) = g(x) • h(x)$ then either $deg((g(x)) = 0$ or $deg((h(x)) = 0$.

There is one more thing which is irreducible element in Integral domains. Let $R$ be a CRU (commutative ring with unity) then an element $ a∈ R $ is called irreducible element, if

(i) $a ≠ 0$ and a is non-unit

(ii) whenever $ a= bc$ for some $b, c ∈ R$ then either $b$ is unit or $c$ is unit in $R$

For example: Consider, $f(x) = 2x^2 + 4∈Z[x] $ then $f(x) = 2(x^2 + 2) = g(x) • h(x)$ (say) Then we saw that, $deg(g(x)) = 0$ so that $f(x)$ is irreducible polynomial in $Z[x]$ but it not an irreducible element in $Z[x]$, since neither $2$ nor $ (x^2 + 2)$ are units in $Z[x]$.

So "irreducible polynomial need not be an irreducible element" But when $F$ is field then "every irreducible polynomial in $F[x]$ is an irreducible element of $F[x]$ and conversely"!

In fact you can check if you take $f(x) = 2x+ 2 ∈ Z_3 [x]$ then $f(x)$ has root in $Z_3$ but $f(x)$ is irreducible polynomial over $Z_3$